Operator norm on the space of continuous multilinear maps

When f is a continuous multilinear map in finitely many variables, we define its norm ‖f‖ as the smallest number such that ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ for all m.

We show that it is indeed a norm, and prove its basic properties.

Main results

Let f be a multilinear map in finitely many variables.

• exists_bound_of_continuous asserts that, if f is continuous, then there exists C > 0 with ‖f m‖ ≤ C * ∏ i, ‖m i‖ for all m.
• continuous_of_bound, conversely, asserts that this bound implies continuity.
• mkContinuous constructs the associated continuous multilinear map.

Let f be a continuous multilinear map in finitely many variables.

• ‖f‖ is its norm, i.e., the smallest number such that ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ for all m.
• le_opNorm f m asserts the fundamental inequality ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖.
• norm_image_sub_le f m₁ m₂ gives a control of the difference f m₁ - f m₂ in terms of ‖f‖ and ‖m₁ - m₂‖.

Implementation notes

We mostly follow the API (and the proofs) of OperatorNorm.lean, with the additional complexity that we should deal with multilinear maps in several variables. The currying/uncurrying constructions are based on those in Multilinear.lean.

From the mathematical point of view, all the results follow from the results on operator norm in one variable, by applying them to one variable after the other through currying. However, this is only well defined when there is an order on the variables (for instance on Fin n) although the final result is independent of the order. While everything could be done following this approach, it turns out that direct proofs are easier and more efficient.

Type variables

We use the following type variables in this file:

• 𝕜 : a NontriviallyNormedField;
• ι, ι' : finite index types with decidable equality;
• E, E₁ : families of normed vector spaces over 𝕜 indexed by i : ι;
• E' : a family of normed vector spaces over 𝕜 indexed by i' : ι';
• Ei : a family of normed vector spaces over 𝕜 indexed by i : Fin (Nat.succ n);
• G, G' : normed vector spaces over 𝕜.

Continuity properties of multilinear maps

We relate continuity of multilinear maps to the inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖, in both directions. Along the way, we prove useful bounds on the difference ‖f m₁ - f m₂‖.

theorem MultilinearMap.norm_map_coord_zero {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf : Continuous ⇑f) {m : (i : ι) → E i} {i : ι} (hi : ‖m i‖ = 0) : ‖f m‖ = 0

If f is a continuous multilinear map in finitely many variables on E and m is an element of ∀ i, E i such that one of the m i has norm 0, then f m has norm 0.

Note that we cannot drop the continuity assumption because f (m : Unit → E) = f (m ()), where the domain has zero norm and the codomain has a nonzero norm does not satisfy this condition.

theorem MultilinearMap.bound_of_shell_of_norm_map_coord_zero {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i) : ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

theorem MultilinearMap.bound_of_shell_of_continuous {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hfc : Continuous ⇑f) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i) : ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

If a continuous multilinear map in finitely many variables on normed spaces satisfies the inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖ on a shell ε i / ‖c i‖ < ‖m i‖ < ε i for some positive numbers ε i and elements c i : 𝕜, 1 < ‖c i‖, then it satisfies this inequality for all m.

theorem MultilinearMap.exists_bound_of_continuous {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (hf : Continuous ⇑f) : ∃ (C : ℝ), 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

If a multilinear map in finitely many variables on normed spaces is continuous, then it satisfies the inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖, for some C which can be chosen to be positive.

theorem MultilinearMap.norm_image_sub_le_of_bound' {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) [DecidableEq ι] {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m₁ : (i : ι) → E i) (m₂ : (i : ι) → E i) : ‖f m₁ - f m₂‖ ≤ C * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

If f satisfies a boundedness property around 0, one can deduce a bound on f m₁ - f m₂ using the multilinearity. Here, we give a precise but hard to use version. See norm_image_sub_le_of_bound for a less precise but more usable version. The bound reads ‖f m - f m'‖ ≤ C * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ‖m 3‖ ‖m' 3‖ * ... * max ‖m n‖ ‖m' n‖ + ..., where the other terms in the sum are the same products where 1 is replaced by any i.

theorem MultilinearMap.norm_image_sub_le_of_bound {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m₁ : (i : ι) → E i) (m₂ : (i : ι) → E i) : ‖f m₁ - f m₂‖ ≤ C * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖

If f satisfies a boundedness property around 0, one can deduce a bound on f m₁ - f m₂ using the multilinearity. Here, we give a usable but not very precise version. See norm_image_sub_le_of_bound' for a more precise but less usable version. The bound is ‖f m - f m'‖ ≤ C * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (card ι - 1).

theorem MultilinearMap.continuous_of_bound {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (C : ℝ) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) : Continuous ⇑f

If a multilinear map satisfies an inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖, then it is continuous.

noncomputable def MultilinearMap.mkContinuous {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (C : ℝ) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) : ContinuousMultilinearMap 𝕜 E G

Constructing a continuous multilinear map from a multilinear map satisfying a boundedness condition.

Equations

• f.mkContinuous C H = { toMultilinearMap := f, cont := ⋯ }

@[simp]

theorem MultilinearMap.coe_mkContinuous {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) (C : ℝ) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) : ⇑(f.mkContinuous C H) = ⇑f

theorem MultilinearMap.restr_norm_le {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k : ℕ} {n : ℕ} (f : MultilinearMap 𝕜 (fun (x : Fin n) => G) G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) {C : ℝ} (H : ∀ (m : Fin n → G), ‖f m‖ ≤ C * ∏ i : Fin n, ‖m i‖) (v : Fin k → G) : ‖(f.restr s hk z) v‖ ≤ C * ‖z‖ ^ (n - k) * ∏ i : Fin k, ‖v i‖

Given a multilinear map in n variables, if one restricts it to k variables putting z on the other coordinates, then the resulting restricted function satisfies an inequality ‖f.restr v‖ ≤ C * ‖z‖^(n-k) * Π ‖v i‖ if the original function satisfies ‖f v‖ ≤ C * Π ‖v i‖.

Continuous multilinear maps

We define the norm ‖f‖ of a continuous multilinear map f in finitely many variables as the smallest number such that ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖ for all m. We show that this defines a normed space structure on ContinuousMultilinearMap 𝕜 E G.

theorem ContinuousMultilinearMap.bound {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ∃ (C : ℝ), 0 < C ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

noncomputable def ContinuousMultilinearMap.opNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ℝ

The operator norm of a continuous multilinear map is the inf of all its bounds.

Equations

• f.opNorm = sInf {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}

noncomputable instance ContinuousMultilinearMap.hasOpNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : Norm (ContinuousMultilinearMap 𝕜 E G)

Equations

• ContinuousMultilinearMap.hasOpNorm = { norm := ContinuousMultilinearMap.opNorm }

noncomputable instance ContinuousMultilinearMap.hasOpNorm' {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] : Norm (ContinuousMultilinearMap 𝕜 (fun (x : ι) => G) G')

An alias of ContinuousMultilinearMap.hasOpNorm with non-dependent types to help typeclass search.

Equations

• ContinuousMultilinearMap.hasOpNorm' = ContinuousMultilinearMap.hasOpNorm

theorem ContinuousMultilinearMap.norm_def {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ‖f‖ = sInf {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}

theorem ContinuousMultilinearMap.bounds_nonempty {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} : ∃ (c : ℝ), c ∈ {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}

theorem ContinuousMultilinearMap.bounds_bddBelow {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} : BddBelow {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖}

theorem ContinuousMultilinearMap.isLeast_opNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : IsLeast {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖} ‖f‖

@[deprecated ContinuousMultilinearMap.isLeast_opNorm]

theorem ContinuousMultilinearMap.isLeast_op_norm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : IsLeast {c : ℝ | 0 ≤ c ∧ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ c * ∏ i : ι, ‖m i‖} ‖f‖

Alias of ContinuousMultilinearMap.isLeast_opNorm.

theorem ContinuousMultilinearMap.opNorm_nonneg {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : 0 ≤ ‖f‖

@[deprecated ContinuousMultilinearMap.opNorm_nonneg]

theorem ContinuousMultilinearMap.op_norm_nonneg {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : 0 ≤ ‖f‖

Alias of ContinuousMultilinearMap.opNorm_nonneg.

theorem ContinuousMultilinearMap.le_opNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖

The fundamental property of the operator norm of a continuous multilinear map: ‖f m‖ is bounded by ‖f‖ times the product of the ‖m i‖.

@[deprecated ContinuousMultilinearMap.le_opNorm]

theorem ContinuousMultilinearMap.le_op_norm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖ ≤ ‖f‖ * ∏ i : ι, ‖m i‖

Alias of ContinuousMultilinearMap.le_opNorm.

---

The fundamental property of the operator norm of a continuous multilinear map: ‖f m‖ is bounded by ‖f‖ times the product of the ‖m i‖.

theorem ContinuousMultilinearMap.le_mul_prod_of_le_opNorm_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} {m : (i : ι) → E i} {C : ℝ} {b : ι → ℝ} (hC : ‖f‖ ≤ C) (hm : ∀ (i : ι), ‖m i‖ ≤ b i) : ‖f m‖ ≤ C * ∏ i : ι, b i

@[deprecated ContinuousMultilinearMap.le_mul_prod_of_le_opNorm_of_le]

theorem ContinuousMultilinearMap.le_mul_prod_of_le_op_norm_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} {m : (i : ι) → E i} {C : ℝ} {b : ι → ℝ} (hC : ‖f‖ ≤ C) (hm : ∀ (i : ι), ‖m i‖ ≤ b i) : ‖f m‖ ≤ C * ∏ i : ι, b i

Alias of ContinuousMultilinearMap.le_mul_prod_of_le_opNorm_of_le.

theorem ContinuousMultilinearMap.le_opNorm_mul_prod_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ι → ℝ} (hm : ∀ (i : ι), ‖m i‖ ≤ b i) : ‖f m‖ ≤ ‖f‖ * ∏ i : ι, b i

@[deprecated ContinuousMultilinearMap.le_opNorm_mul_prod_of_le]

theorem ContinuousMultilinearMap.le_op_norm_mul_prod_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ι → ℝ} (hm : ∀ (i : ι), ‖m i‖ ≤ b i) : ‖f m‖ ≤ ‖f‖ * ∏ i : ι, b i

Alias of ContinuousMultilinearMap.le_opNorm_mul_prod_of_le.

theorem ContinuousMultilinearMap.le_opNorm_mul_pow_card_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ℝ} (hm : ‖m‖ ≤ b) : ‖f m‖ ≤ ‖f‖ * b ^ Fintype.card ι

@[deprecated ContinuousMultilinearMap.le_opNorm_mul_pow_card_of_le]

theorem ContinuousMultilinearMap.le_op_norm_mul_pow_card_of_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {m : (i : ι) → E i} {b : ℝ} (hm : ‖m‖ ≤ b) : ‖f m‖ ≤ ‖f‖ * b ^ Fintype.card ι

Alias of ContinuousMultilinearMap.le_opNorm_mul_pow_card_of_le.

theorem ContinuousMultilinearMap.le_opNorm_mul_pow_of_le {𝕜 : Type u} {G : Type wG} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} {Ei : Fin n → Type u_1} [(i : Fin n) → SeminormedAddCommGroup (Ei i)] [(i : Fin n) → NormedSpace 𝕜 (Ei i)] (f : ContinuousMultilinearMap 𝕜 Ei G) {m : (i : Fin n) → Ei i} {b : ℝ} (hm : ‖m‖ ≤ b) : ‖f m‖ ≤ ‖f‖ * b ^ n

@[deprecated ContinuousMultilinearMap.le_opNorm_mul_pow_of_le]

theorem ContinuousMultilinearMap.le_op_norm_mul_pow_of_le {𝕜 : Type u} {G : Type wG} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {n : ℕ} {Ei : Fin n → Type u_1} [(i : Fin n) → SeminormedAddCommGroup (Ei i)] [(i : Fin n) → NormedSpace 𝕜 (Ei i)] (f : ContinuousMultilinearMap 𝕜 Ei G) {m : (i : Fin n) → Ei i} {b : ℝ} (hm : ‖m‖ ≤ b) : ‖f m‖ ≤ ‖f‖ * b ^ n

Alias of ContinuousMultilinearMap.le_opNorm_mul_pow_of_le.

theorem ContinuousMultilinearMap.le_of_opNorm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} (m : (i : ι) → E i) {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

@[deprecated ContinuousMultilinearMap.le_of_opNorm_le]

theorem ContinuousMultilinearMap.le_of_op_norm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {f : ContinuousMultilinearMap 𝕜 E G} (m : (i : ι) → E i) {C : ℝ} (h : ‖f‖ ≤ C) : ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

Alias of ContinuousMultilinearMap.le_of_opNorm_le.

theorem ContinuousMultilinearMap.ratio_le_opNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖ / ∏ i : ι, ‖m i‖ ≤ ‖f‖

@[deprecated ContinuousMultilinearMap.ratio_le_opNorm]

theorem ContinuousMultilinearMap.ratio_le_op_norm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖ / ∏ i : ι, ‖m i‖ ≤ ‖f‖

Alias of ContinuousMultilinearMap.ratio_le_opNorm.

theorem ContinuousMultilinearMap.unit_le_opNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖

The image of the unit ball under a continuous multilinear map is bounded.

@[deprecated ContinuousMultilinearMap.unit_le_opNorm]

theorem ContinuousMultilinearMap.unit_le_op_norm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) (h : ‖m‖ ≤ 1) : ‖f m‖ ≤ ‖f‖

Alias of ContinuousMultilinearMap.unit_le_opNorm.

---

The image of the unit ball under a continuous multilinear map is bounded.

theorem ContinuousMultilinearMap.opNorm_le_bound {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ M * ∏ i : ι, ‖m i‖) : ‖f‖ ≤ M

If one controls the norm of every f x, then one controls the norm of f.

@[deprecated ContinuousMultilinearMap.opNorm_le_bound]

theorem ContinuousMultilinearMap.op_norm_le_bound {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {M : ℝ} (hMp : 0 ≤ M) (hM : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ M * ∏ i : ι, ‖m i‖) : ‖f‖ ≤ M

Alias of ContinuousMultilinearMap.opNorm_le_bound.

---

If one controls the norm of every f x, then one controls the norm of f.

theorem ContinuousMultilinearMap.opNorm_le_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) : ‖f‖ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

@[deprecated ContinuousMultilinearMap.opNorm_le_iff]

theorem ContinuousMultilinearMap.op_norm_le_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) : ‖f‖ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

Alias of ContinuousMultilinearMap.opNorm_le_iff.

theorem ContinuousMultilinearMap.opNorm_add_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G) : ‖f + g‖ ≤ ‖f‖ + ‖g‖

The operator norm satisfies the triangle inequality.

@[deprecated ContinuousMultilinearMap.opNorm_add_le]

theorem ContinuousMultilinearMap.op_norm_add_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G) : ‖f + g‖ ≤ ‖f‖ + ‖g‖

Alias of ContinuousMultilinearMap.opNorm_add_le.

---

The operator norm satisfies the triangle inequality.

theorem ContinuousMultilinearMap.opNorm_zero {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : ‖0‖ = 0

@[deprecated ContinuousMultilinearMap.opNorm_zero]

theorem ContinuousMultilinearMap.op_norm_zero {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : ‖0‖ = 0

Alias of ContinuousMultilinearMap.opNorm_zero.

theorem ContinuousMultilinearMap.opNorm_smul_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {𝕜' : Type u_1} [NormedField 𝕜'] [NormedSpace 𝕜' G] [SMulCommClass 𝕜 𝕜' G] (c : 𝕜') : ‖c • f‖ ≤ ‖c‖ * ‖f‖

@[deprecated ContinuousMultilinearMap.opNorm_smul_le]

theorem ContinuousMultilinearMap.op_norm_smul_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {𝕜' : Type u_1} [NormedField 𝕜'] [NormedSpace 𝕜' G] [SMulCommClass 𝕜 𝕜' G] (c : 𝕜') : ‖c • f‖ ≤ ‖c‖ * ‖f‖

Alias of ContinuousMultilinearMap.opNorm_smul_le.

theorem ContinuousMultilinearMap.opNorm_neg {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ‖-f‖ = ‖f‖

@[deprecated ContinuousMultilinearMap.opNorm_neg]

theorem ContinuousMultilinearMap.op_norm_neg {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ‖-f‖ = ‖f‖

Alias of ContinuousMultilinearMap.opNorm_neg.

noncomputable def ContinuousMultilinearMap.seminorm (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : Seminorm 𝕜 (ContinuousMultilinearMap 𝕜 E G)

Operator seminorm on the space of continuous multilinear maps, as Seminorm.

We use this seminorm to define a SeminormedAddCommGroup structure on ContinuousMultilinearMap 𝕜 E G, but we have to override the projection UniformSpace so that it is definitionally equal to the one coming from the topologies on E and G.

Equations

• ContinuousMultilinearMap.seminorm 𝕜 E G = Seminorm.ofSMulLE norm ⋯ ⋯ ⋯

noncomputable instance ContinuousMultilinearMap.instPseudoMetricSpace {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : PseudoMetricSpace (ContinuousMultilinearMap 𝕜 E G)

Equations

• ContinuousMultilinearMap.instPseudoMetricSpace = SeminormedAddCommGroup.toPseudoMetricSpace.replaceUniformity ⋯

noncomputable instance ContinuousMultilinearMap.seminormedAddCommGroup {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : SeminormedAddCommGroup (ContinuousMultilinearMap 𝕜 E G)

Continuous multilinear maps themselves form a seminormed space with respect to the operator norm.

Equations

• ContinuousMultilinearMap.seminormedAddCommGroup = SeminormedAddCommGroup.mk ⋯

noncomputable instance ContinuousMultilinearMap.seminormedAddCommGroup' {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] : SeminormedAddCommGroup (ContinuousMultilinearMap 𝕜 (fun (x : ι) => G) G')

An alias of ContinuousMultilinearMap.seminormedAddCommGroup with non-dependent types to help typeclass search.

Equations

• ContinuousMultilinearMap.seminormedAddCommGroup' = ContinuousMultilinearMap.seminormedAddCommGroup

noncomputable instance ContinuousMultilinearMap.normedSpace {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {𝕜' : Type u_1} [NormedField 𝕜'] [NormedSpace 𝕜' G] [SMulCommClass 𝕜 𝕜' G] : NormedSpace 𝕜' (ContinuousMultilinearMap 𝕜 E G)

Equations

• ContinuousMultilinearMap.normedSpace = NormedSpace.mk ⋯

noncomputable instance ContinuousMultilinearMap.normedSpace' {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {𝕜' : Type u_1} [NormedField 𝕜'] [NormedSpace 𝕜' G] [SMulCommClass 𝕜 𝕜' G] : NormedSpace 𝕜' (ContinuousMultilinearMap 𝕜 (fun (x : ι) => G') G)

An alias of ContinuousMultilinearMap.normedSpace with non-dependent types to help typeclass search.

Equations

• ContinuousMultilinearMap.normedSpace' = ContinuousMultilinearMap.normedSpace

theorem ContinuousMultilinearMap.le_opNNNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖₊ ≤ ‖f‖₊ * ∏ i : ι, ‖m i‖₊

The fundamental property of the operator norm of a continuous multilinear map: ‖f m‖ is bounded by ‖f‖ times the product of the ‖m i‖, nnnorm version.

@[deprecated ContinuousMultilinearMap.le_opNNNorm]

theorem ContinuousMultilinearMap.le_op_nnnorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) : ‖f m‖₊ ≤ ‖f‖₊ * ∏ i : ι, ‖m i‖₊

Alias of ContinuousMultilinearMap.le_opNNNorm.

---

The fundamental property of the operator norm of a continuous multilinear map: ‖f m‖ is bounded by ‖f‖ times the product of the ‖m i‖, nnnorm version.

theorem ContinuousMultilinearMap.le_of_opNNNorm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) {C : NNReal} (h : ‖f‖₊ ≤ C) : ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊

@[deprecated ContinuousMultilinearMap.le_of_opNNNorm_le]

theorem ContinuousMultilinearMap.le_of_op_nnnorm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m : (i : ι) → E i) {C : NNReal} (h : ‖f‖₊ ≤ C) : ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊

Alias of ContinuousMultilinearMap.le_of_opNNNorm_le.

theorem ContinuousMultilinearMap.opNNNorm_le_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : NNReal} : ‖f‖₊ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊

@[deprecated ContinuousMultilinearMap.opNNNorm_le_iff]

theorem ContinuousMultilinearMap.op_nnnorm_le_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {C : NNReal} : ‖f‖₊ ≤ C ↔ ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊

Alias of ContinuousMultilinearMap.opNNNorm_le_iff.

theorem ContinuousMultilinearMap.isLeast_opNNNorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : IsLeast {C : NNReal | ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊} ‖f‖₊

@[deprecated ContinuousMultilinearMap.isLeast_opNNNorm]

theorem ContinuousMultilinearMap.isLeast_op_nnnorm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : IsLeast {C : NNReal | ∀ (m : (i : ι) → E i), ‖f m‖₊ ≤ C * ∏ i : ι, ‖m i‖₊} ‖f‖₊

Alias of ContinuousMultilinearMap.isLeast_opNNNorm.

theorem ContinuousMultilinearMap.opNNNorm_prod {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') : ‖f.prod g‖₊ = max ‖f‖₊ ‖g‖₊

@[deprecated ContinuousMultilinearMap.opNNNorm_prod]

theorem ContinuousMultilinearMap.op_nnnorm_prod {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') : ‖f.prod g‖₊ = max ‖f‖₊ ‖g‖₊

Alias of ContinuousMultilinearMap.opNNNorm_prod.

theorem ContinuousMultilinearMap.opNorm_prod {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') : ‖f.prod g‖ = max ‖f‖ ‖g‖

@[deprecated ContinuousMultilinearMap.opNorm_prod]

theorem ContinuousMultilinearMap.op_norm_prod {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : ContinuousMultilinearMap 𝕜 E G) (g : ContinuousMultilinearMap 𝕜 E G') : ‖f.prod g‖ = max ‖f‖ ‖g‖

Alias of ContinuousMultilinearMap.opNorm_prod.

theorem ContinuousMultilinearMap.opNNNorm_pi {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i' : ι') → SeminormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] (f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')) : ‖ContinuousMultilinearMap.pi f‖₊ = ‖f‖₊

theorem ContinuousMultilinearMap.opNorm_pi {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'} [(i' : ι') → SeminormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] (f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')) : ‖ContinuousMultilinearMap.pi f‖ = ‖f‖

@[deprecated ContinuousMultilinearMap.opNorm_pi]

theorem ContinuousMultilinearMap.op_norm_pi {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'} [(i' : ι') → SeminormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] (f : (i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')) : ‖ContinuousMultilinearMap.pi f‖ = ‖f‖

Alias of ContinuousMultilinearMap.opNorm_pi.

@[simp]

theorem ContinuousMultilinearMap.norm_ofSubsingleton {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) (f : G →L[𝕜] G') : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖

@[simp]

theorem ContinuousMultilinearMap.nnnorm_ofSubsingleton {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) (f : G →L[𝕜] G') : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i) f‖₊ = ‖f‖₊

noncomputable def ContinuousMultilinearMap.ofSubsingletonₗᵢ (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) : (G →L[𝕜] G') ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun (x : ι) => G) G'

Linear isometry between continuous linear maps from G to G' and continuous 1-multilinear maps from G to G'.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousMultilinearMap.ofSubsingletonₗᵢ_symm_apply (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) : ∀ (a : ContinuousMultilinearMap 𝕜 (fun (x : ι) => G) G'), (ContinuousMultilinearMap.ofSubsingletonₗᵢ 𝕜 G i).symm a = (ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i).invFun a

@[simp]

theorem ContinuousMultilinearMap.ofSubsingletonₗᵢ_apply (𝕜 : Type u) {ι : Type v} (G : Type wG) {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] [Subsingleton ι] (i : ι) : ∀ (a : G →L[𝕜] G'), (ContinuousMultilinearMap.ofSubsingletonₗᵢ 𝕜 G i) a = (ContinuousMultilinearMap.ofSubsingleton 𝕜 G G' i).toFun a

theorem ContinuousMultilinearMap.norm_ofSubsingleton_id_le (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] (i : ι) : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ ≤ 1

theorem ContinuousMultilinearMap.nnnorm_ofSubsingleton_id_le (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] (i : ι) : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖₊ ≤ 1

@[simp]

theorem ContinuousMultilinearMap.norm_constOfIsEmpty (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [IsEmpty ι] (x : G) : ‖ContinuousMultilinearMap.constOfIsEmpty 𝕜 E x‖ = ‖x‖

@[simp]

theorem ContinuousMultilinearMap.nnnorm_constOfIsEmpty (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [IsEmpty ι] (x : G) : ‖ContinuousMultilinearMap.constOfIsEmpty 𝕜 E x‖₊ = ‖x‖₊

noncomputable def ContinuousMultilinearMap.prodL (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (G : Type wG) (G' : Type wG') [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G' ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 E (G × G')

ContinuousMultilinearMap.prod as a LinearIsometryEquiv.

Equations

• One or more equations did not get rendered due to their size.

noncomputable def ContinuousMultilinearMap.piₗᵢ (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] {ι' : Type v'} [Fintype ι'] {E' : ι' → Type wE'} [(i' : ι') → NormedAddCommGroup (E' i')] [(i' : ι') → NormedSpace 𝕜 (E' i')] : ((i' : ι') → ContinuousMultilinearMap 𝕜 E (E' i')) ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 E ((i : ι') → E' i)

ContinuousMultilinearMap.pi as a LinearIsometryEquiv.

Equations

• ContinuousMultilinearMap.piₗᵢ 𝕜 E = { toLinearEquiv := ContinuousMultilinearMap.piLinearEquiv, norm_map' := ⋯ }

@[simp]

theorem ContinuousMultilinearMap.norm_restrictScalars {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)] : ‖ContinuousMultilinearMap.restrictScalars 𝕜' f‖ = ‖f‖

noncomputable def ContinuousMultilinearMap.restrictScalarsₗᵢ {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (𝕜' : Type u_1) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)] : ContinuousMultilinearMap 𝕜 E G →ₗᵢ[𝕜'] ContinuousMultilinearMap 𝕜' E G

ContinuousMultilinearMap.restrictScalars as a LinearIsometry.

Equations

• ContinuousMultilinearMap.restrictScalarsₗᵢ 𝕜' = { toFun := ContinuousMultilinearMap.restrictScalars 𝕜', map_add' := ⋯, map_smul' := ⋯, norm_map' := ⋯ }

noncomputable def ContinuousMultilinearMap.restrictScalarsLinear {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (𝕜' : Type u_1) [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)] : ContinuousMultilinearMap 𝕜 E G →L[𝕜'] ContinuousMultilinearMap 𝕜' E G

ContinuousMultilinearMap.restrictScalars as a ContinuousLinearMap.

Equations

• ContinuousMultilinearMap.restrictScalarsLinear 𝕜' = (ContinuousMultilinearMap.restrictScalarsₗᵢ 𝕜').toContinuousLinearMap

theorem ContinuousMultilinearMap.continuous_restrictScalars {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {𝕜' : Type u_1} [NontriviallyNormedField 𝕜'] [NormedAlgebra 𝕜' 𝕜] [NormedSpace 𝕜' G] [IsScalarTower 𝕜' 𝕜 G] [(i : ι) → NormedSpace 𝕜' (E i)] [∀ (i : ι), IsScalarTower 𝕜' 𝕜 (E i)] : Continuous (ContinuousMultilinearMap.restrictScalars 𝕜')

theorem ContinuousMultilinearMap.norm_image_sub_le' {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) [DecidableEq ι] (m₁ : (i : ι) → E i) (m₂ : (i : ι) → E i) : ‖f m₁ - f m₂‖ ≤ ‖f‖ * ∑ i : ι, ∏ j : ι, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

The difference f m₁ - f m₂ is controlled in terms of ‖f‖ and ‖m₁ - m₂‖, precise version. For a less precise but more usable version, see norm_image_sub_le. The bound reads ‖f m - f m'‖ ≤ ‖f‖ * ‖m 1 - m' 1‖ * max ‖m 2‖ ‖m' 2‖ * max ‖m 3‖ ‖m' 3‖ * ... * max ‖m n‖ ‖m' n‖ + ..., where the other terms in the sum are the same products where 1 is replaced by any i.

theorem ContinuousMultilinearMap.norm_image_sub_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) (m₁ : (i : ι) → E i) (m₂ : (i : ι) → E i) : ‖f m₁ - f m₂‖ ≤ ‖f‖ * ↑(Fintype.card ι) * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1) * ‖m₁ - m₂‖

The difference f m₁ - f m₂ is controlled in terms of ‖f‖ and ‖m₁ - m₂‖, less precise version. For a more precise but less usable version, see norm_image_sub_le'. The bound is ‖f m - f m'‖ ≤ ‖f‖ * card ι * ‖m - m'‖ * (max ‖m‖ ‖m'‖) ^ (card ι - 1).

theorem ContinuousMultilinearMap.continuous_eval {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : Continuous fun (p : ContinuousMultilinearMap 𝕜 E G × ((i : ι) → E i)) => p.1 p.2

Applying a multilinear map to a vector is continuous in both coordinates.

theorem MultilinearMap.mkContinuous_norm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) : ‖f.mkContinuous C H‖ ≤ C

If a continuous multilinear map is constructed from a multilinear map via the constructor mkContinuous, then its norm is bounded by the bound given to the constructor if it is nonnegative.

theorem MultilinearMap.mkContinuous_norm_le' {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {C : ℝ} (H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) : ‖f.mkContinuous C H‖ ≤ max C 0

If a continuous multilinear map is constructed from a multilinear map via the constructor mkContinuous, then its norm is bounded by the bound given to the constructor if it is nonnegative.

noncomputable def ContinuousMultilinearMap.restr {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k : ℕ} {n : ℕ} (f : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) => G) G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) : ContinuousMultilinearMap 𝕜 (fun (i : Fin k) => G) G'

Given a continuous multilinear map f on n variables (parameterized by Fin n) and a subset s of k of these variables, one gets a new continuous multilinear map on Fin k by varying these variables, and fixing the other ones equal to a given value z. It is denoted by f.restr s hk z, where hk is a proof that the cardinality of s is k. The implicit identification between Fin k and s that we use is the canonical (increasing) bijection.

Equations

• f.restr s hk z = (f.restr s hk z).mkContinuous (‖f‖ * ‖z‖ ^ (n - k)) ⋯

theorem ContinuousMultilinearMap.norm_restr {𝕜 : Type u} {G : Type wG} {G' : Type wG'} [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] {k : ℕ} {n : ℕ} (f : ContinuousMultilinearMap 𝕜 (fun (i : Fin n) => G) G') (s : Finset (Fin n)) (hk : s.card = k) (z : G) : ‖f.restr s hk z‖ ≤ ‖f‖ * ‖z‖ ^ (n - k)

@[simp]

theorem ContinuousMultilinearMap.norm_mkPiAlgebra_le {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [Nonempty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ 1

theorem ContinuousMultilinearMap.norm_mkPiAlgebra_of_empty {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [IsEmpty ι] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = ‖1‖

@[simp]

theorem ContinuousMultilinearMap.norm_mkPiAlgebra {𝕜 : Type u} {ι : Type v} [Fintype ι] [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedCommRing A] [NormedAlgebra 𝕜 A] [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ = 1

theorem ContinuousMultilinearMap.norm_mkPiAlgebraFin_succ_le {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n.succ A‖ ≤ 1

theorem ContinuousMultilinearMap.norm_mkPiAlgebraFin_le_of_pos {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] (hn : 0 < n) : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ ≤ 1

theorem ContinuousMultilinearMap.norm_mkPiAlgebraFin_zero {𝕜 : Type u} [NontriviallyNormedField 𝕜] {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖1‖

@[simp]

theorem ContinuousMultilinearMap.norm_mkPiAlgebraFin {𝕜 : Type u} [NontriviallyNormedField 𝕜] {n : ℕ} {A : Type u_1} [NormedRing A] [NormedAlgebra 𝕜 A] [NormOneClass A] : ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 n A‖ = 1

@[simp]

theorem ContinuousMultilinearMap.nnnorm_smulRight {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) : ‖f.smulRight z‖₊ = ‖f‖₊ * ‖z‖₊

@[simp]

theorem ContinuousMultilinearMap.norm_smulRight {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) : ‖f.smulRight z‖ = ‖f‖ * ‖z‖

@[simp]

theorem ContinuousMultilinearMap.norm_mkPiRing {𝕜 : Type u} {ι : Type v} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (z : G) : ‖ContinuousMultilinearMap.mkPiRing 𝕜 ι z‖ = ‖z‖

noncomputable def ContinuousMultilinearMap.smulRightL (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : ContinuousMultilinearMap 𝕜 E 𝕜 →L[𝕜] G →L[𝕜] ContinuousMultilinearMap 𝕜 E G

Continuous bilinear map realizing (f, z) ↦ f.smulRight z.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousMultilinearMap.smulRightL_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E 𝕜) (z : G) : ((ContinuousMultilinearMap.smulRightL 𝕜 E G) f) z = f.smulRight z

theorem ContinuousMultilinearMap.norm_smulRightL_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : ‖ContinuousMultilinearMap.smulRightL 𝕜 E G‖ ≤ 1

noncomputable def ContinuousMultilinearMap.piFieldEquiv (𝕜 : Type u) (ι : Type v) (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : G ≃ₗᵢ[𝕜] ContinuousMultilinearMap 𝕜 (fun (x : ι) => 𝕜) G

Continuous multilinear maps on 𝕜^n with values in G are in bijection with G, as such a continuous multilinear map is completely determined by its value on the constant vector made of ones. We register this bijection as a linear isometry in ContinuousMultilinearMap.piFieldEquiv.

Equations

• One or more equations did not get rendered due to their size.

theorem ContinuousLinearMap.norm_compContinuousMultilinearMap_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G →L[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G) : ‖g.compContinuousMultilinearMap f‖ ≤ ‖g‖ * ‖f‖

noncomputable def ContinuousLinearMap.compContinuousMultilinearMapL (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (G : Type wG) (G' : Type wG') [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] : (G →L[𝕜] G') →L[𝕜] ContinuousMultilinearMap 𝕜 E G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'

ContinuousLinearMap.compContinuousMultilinearMap as a bundled continuous bilinear map.

Equations

• ContinuousLinearMap.compContinuousMultilinearMapL 𝕜 E G G' = (LinearMap.mk₂ 𝕜 ContinuousLinearMap.compContinuousMultilinearMap ⋯ ⋯ ⋯ ⋯).mkContinuous₂ 1 ⋯

noncomputable def ContinuousLinearEquiv.compContinuousMultilinearMapL {𝕜 : Type u} {ι : Type v} (E : ι → Type wE) {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G ≃L[𝕜] G') : ContinuousMultilinearMap 𝕜 E G ≃L[𝕜] ContinuousMultilinearMap 𝕜 E G'

ContinuousLinearMap.compContinuousMultilinearMap as a bundled continuous linear equiv.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousLinearEquiv.compContinuousMultilinearMapL_symm {𝕜 : Type u} {ι : Type v} (E : ι → Type wE) {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G ≃L[𝕜] G') : (ContinuousLinearEquiv.compContinuousMultilinearMapL E g).symm = ContinuousLinearEquiv.compContinuousMultilinearMapL E g.symm

@[simp]

theorem ContinuousLinearEquiv.compContinuousMultilinearMapL_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G ≃L[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G) : (ContinuousLinearEquiv.compContinuousMultilinearMapL E g) f = (↑g).compContinuousMultilinearMap f

noncomputable def ContinuousLinearMap.flipMultilinear {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')

Flip arguments in f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G' to get ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousLinearMap.flipMultilinear_apply_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') (m : (i : ι) → E i) (x : G) : (f.flipMultilinear m) x = (f x) m

theorem LinearIsometry.norm_compContinuousMultilinearMap {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (g : G →ₗᵢ[𝕜] G') (f : ContinuousMultilinearMap 𝕜 E G) : ‖g.toContinuousLinearMap.compContinuousMultilinearMap f‖ = ‖f‖

noncomputable def MultilinearMap.mkContinuousLinear {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') (C : ℝ) (H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖) : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'

Given a map f : G →ₗ[𝕜] MultilinearMap 𝕜 E G' and an estimate H : ∀ x m, ‖f x m‖ ≤ C * ‖x‖ * ∏ i, ‖m i‖, construct a continuous linear map from G to ContinuousMultilinearMap 𝕜 E G'.

In order to lift, e.g., a map f : (MultilinearMap 𝕜 E G) →ₗ[𝕜] MultilinearMap 𝕜 E' G' to a map (ContinuousMultilinearMap 𝕜 E G) →L[𝕜] ContinuousMultilinearMap 𝕜 E' G', one can apply this construction to f.comp ContinuousMultilinearMap.toMultilinearMapLinear which is a linear map from ContinuousMultilinearMap 𝕜 E G to MultilinearMap 𝕜 E' G'.

Equations

• MultilinearMap.mkContinuousLinear f C H = { toFun := fun (x : G) => (f x).mkContinuous (C * ‖x‖) ⋯, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (max C 0) ⋯

theorem MultilinearMap.mkContinuousLinear_norm_le' {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') (C : ℝ) (H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖) : ‖MultilinearMap.mkContinuousLinear f C H‖ ≤ max C 0

theorem MultilinearMap.mkContinuousLinear_norm_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] (f : G →ₗ[𝕜] MultilinearMap 𝕜 E G') {C : ℝ} (hC : 0 ≤ C) (H : ∀ (x : G) (m : (i : ι) → E i), ‖(f x) m‖ ≤ C * ‖x‖ * ∏ i : ι, ‖m i‖) : ‖MultilinearMap.mkContinuousLinear f C H‖ ≤ C

noncomputable def MultilinearMap.mkContinuousMultilinear {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) (C : ℝ) (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖) : ContinuousMultilinearMap 𝕜 E (ContinuousMultilinearMap 𝕜 E' G)

Given a map f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G) and an estimate H : ∀ m m', ‖f m m'‖ ≤ C * ∏ i, ‖m i‖ * ∏ i, ‖m' i‖, upgrade all MultilinearMaps in the type to ContinuousMultilinearMaps.

Equations

• f.mkContinuousMultilinear C H = { toFun := fun (m : (i : ι) → E i) => (f m).mkContinuous (C * ∏ i : ι, ‖m i‖) ⋯, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous (max C 0) ⋯

@[simp]

theorem MultilinearMap.mkContinuousMultilinear_apply {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ} (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖) (m : (i : ι) → E i) : ⇑((f.mkContinuousMultilinear C H) m) = ⇑(f m)

theorem MultilinearMap.mkContinuousMultilinear_norm_le' {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) (C : ℝ) (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖) : ‖f.mkContinuousMultilinear C H‖ ≤ max C 0

theorem MultilinearMap.mkContinuousMultilinear_norm_le {𝕜 : Type u} {ι : Type v} {ι' : Type v'} {E : ι → Type wE} {E' : ι' → Type wE'} {G : Type wG} [Fintype ι] [Fintype ι'] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι') → SeminormedAddCommGroup (E' i)] [(i : ι') → NormedSpace 𝕜 (E' i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E (MultilinearMap 𝕜 E' G)) {C : ℝ} (hC : 0 ≤ C) (H : ∀ (m₁ : (i : ι) → E i) (m₂ : (i : ι') → E' i), ‖(f m₁) m₂‖ ≤ (C * ∏ i : ι, ‖m₁ i‖) * ∏ i : ι', ‖m₂ i‖) : ‖f.mkContinuousMultilinear C H‖ ≤ C

theorem ContinuousMultilinearMap.norm_compContinuousLinearMap_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i) : ‖g.compContinuousLinearMap f‖ ≤ ‖g‖ * ∏ i : ι, ‖f i‖

theorem ContinuousMultilinearMap.norm_compContinuous_linearIsometry_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →ₗᵢ[𝕜] E₁ i) : ‖g.compContinuousLinearMap fun (i : ι) => (f i).toContinuousLinearMap‖ ≤ ‖g‖

theorem ContinuousMultilinearMap.norm_compContinuous_linearIsometryEquiv {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i) : ‖g.compContinuousLinearMap fun (i : ι) => ↑{ toLinearEquiv := (f i).toLinearEquiv, continuous_toFun := ⋯, continuous_invFun := ⋯ }‖ = ‖g‖

noncomputable def ContinuousMultilinearMap.compContinuousLinearMapL {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i →L[𝕜] E₁ i) : ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G

ContinuousMultilinearMap.compContinuousLinearMap as a bundled continuous linear map. This implementation fixes f : Π i, E i →L[𝕜] E₁ i.

Actually, the map is multilinear in f, see ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear.

For a version fixing g and varying f, see compContinuousLinearMapLRight.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousMultilinearMap.compContinuousLinearMapL_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i) : (ContinuousMultilinearMap.compContinuousLinearMapL f) g = g.compContinuousLinearMap f

theorem ContinuousMultilinearMap.norm_compContinuousLinearMapL_le {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i →L[𝕜] E₁ i) : ‖ContinuousMultilinearMap.compContinuousLinearMapL f‖ ≤ ∏ i : ι, ‖f i‖

noncomputable def ContinuousMultilinearMap.compContinuousLinearMapLRight {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) : ContinuousMultilinearMap 𝕜 (fun (i : ι) => E i →L[𝕜] E₁ i) (ContinuousMultilinearMap 𝕜 E G)

ContinuousMultilinearMap.compContinuousLinearMap as a bundled continuous linear map. This implementation fixes g : ContinuousMultilinearMap 𝕜 E₁ G.

Actually, the map is linear in g, see ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear.

For a version fixing f and varying g, see compContinuousLinearMapL.

Equations

• g.compContinuousLinearMapLRight = { toFun := fun (f : (i : ι) → E i →L[𝕜] E₁ i) => g.compContinuousLinearMap f, map_add' := ⋯, map_smul' := ⋯ }.mkContinuous ‖g‖ ⋯

@[simp]

theorem ContinuousMultilinearMap.compContinuousLinearMapLRight_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i →L[𝕜] E₁ i) : g.compContinuousLinearMapLRight f = g.compContinuousLinearMap f

theorem ContinuousMultilinearMap.norm_compContinuousLinearMapLRight_le {𝕜 : Type u} {ι : Type v} (E : ι → Type wE) {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) : ‖g.compContinuousLinearMapLRight‖ ≤ ‖g‖

noncomputable def ContinuousMultilinearMap.compContinuousLinearMapMultilinear (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (E₁ : ι → Type wE₁) (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : MultilinearMap 𝕜 (fun (i : ι) => E i →L[𝕜] E₁ i) (ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)

If f is a collection of continuous linear maps, then the construction ContinuousMultilinearMap.compContinuousLinearMap sending a continuous multilinear map g to g (f₁ ·, ..., fₙ ·) is continuous-linear in g and multilinear in f₁, ..., fₙ.

Equations

• ContinuousMultilinearMap.compContinuousLinearMapMultilinear 𝕜 E E₁ G = { toFun := ContinuousMultilinearMap.compContinuousLinearMapL, map_add' := ⋯, map_smul' := ⋯ }

noncomputable def ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear (𝕜 : Type u) {ι : Type v} (E : ι → Type wE) (E₁ : ι → Type wE₁) (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] : ContinuousMultilinearMap 𝕜 (fun (i : ι) => E i →L[𝕜] E₁ i) (ContinuousMultilinearMap 𝕜 E₁ G →L[𝕜] ContinuousMultilinearMap 𝕜 E G)

If f is a collection of continuous linear maps, then the construction ContinuousMultilinearMap.compContinuousLinearMap sending a continuous multilinear map g to g (f₁ ·, ..., fₙ ·) is continuous-linear in g and continuous-multilinear in f₁, ..., fₙ.

Equations

• ContinuousMultilinearMap.compContinuousLinearMapContinuousMultilinear 𝕜 E E₁ G = (ContinuousMultilinearMap.compContinuousLinearMapMultilinear 𝕜 E E₁ G).mkContinuous 1 ⋯

noncomputable def ContinuousMultilinearMap.compContinuousLinearMapEquivL {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i ≃L[𝕜] E₁ i) : ContinuousMultilinearMap 𝕜 E₁ G ≃L[𝕜] ContinuousMultilinearMap 𝕜 E G

ContinuousMultilinearMap.compContinuousLinearMap as a bundled continuous linear equiv, given f : Π i, E i ≃L[𝕜] E₁ i.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem ContinuousMultilinearMap.compContinuousLinearMapEquivL_symm {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : (i : ι) → E i ≃L[𝕜] E₁ i) : (ContinuousMultilinearMap.compContinuousLinearMapEquivL G f).symm = ContinuousMultilinearMap.compContinuousLinearMapEquivL G fun (i : ι) => (f i).symm

@[simp]

theorem ContinuousMultilinearMap.compContinuousLinearMapEquivL_apply {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (g : ContinuousMultilinearMap 𝕜 E₁ G) (f : (i : ι) → E i ≃L[𝕜] E₁ i) : (ContinuousMultilinearMap.compContinuousLinearMapEquivL G f) g = g.compContinuousLinearMap fun (i : ι) => ↑(f i)

noncomputable def ContinuousMultilinearMap.iteratedFDerivComponent {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {α : Type u_1} [Fintype α] [DecidableEq ι] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s) [DecidablePred fun (x : ι) => x ∈ s] : ContinuousMultilinearMap 𝕜 (fun (i : { a : ι // a ∉ s }) => E₁ ↑i) (ContinuousMultilinearMap 𝕜 (fun (x : α) => (i : ι) → E₁ i) G)

One of the components of the iterated derivative of a continuous multilinear map. Given a bijection e between a type α (typically Fin k) and a subset s of ι, this component is a continuous multilinear map of k vectors v₁, ..., vₖ, mapping them to f (x₁, (v_{e.symm 2})₂, x₃, ...), where at indices i in s one uses the i-th coordinate of the vector v_{e.symm i} and otherwise one uses the i-th coordinate of a reference vector x. This is continuous multilinear in the components of x outside of s, and in the v_j.

Equations

• f.iteratedFDerivComponent e = (f.iteratedFDerivComponent e).mkContinuousMultilinear ‖f‖ ⋯

@[simp]

theorem ContinuousMultilinearMap.iteratedFDerivComponent_apply {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {α : Type u_1} [Fintype α] [DecidableEq ι] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s) [DecidablePred fun (x : ι) => x ∈ s] (v : (i : { a : ι // a ∉ s }) → E₁ ↑i) (w : α → (i : ι) → E₁ i) : ((f.iteratedFDerivComponent e) v) w = f fun (j : ι) => if h : j ∈ s then w (e.symm ⟨j, h⟩) j else v ⟨j, h⟩

theorem ContinuousMultilinearMap.norm_iteratedFDerivComponent_le {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] {α : Type u_1} [Fintype α] [DecidableEq ι] (f : ContinuousMultilinearMap 𝕜 E₁ G) {s : Set ι} (e : α ≃ ↑s) [DecidablePred fun (x : ι) => x ∈ s] (x : (i : ι) → E₁ i) : ‖(f.iteratedFDerivComponent e) fun (x_1 : { a : ι // a ∉ s }) => x ↑x_1‖ ≤ ‖f‖ * ‖x‖ ^ (Fintype.card ι - Fintype.card α)

noncomputable def ContinuousMultilinearMap.iteratedFDeriv {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E₁ G) (k : ℕ) (x : (i : ι) → E₁ i) : ContinuousMultilinearMap 𝕜 (fun (x : Fin k) => (i : ι) → E₁ i) G

The k-th iterated derivative of a continuous multilinear map f at the point x. It is a continuous multilinear map of k vectors v₁, ..., vₖ (with the same type as x), mapping them to ∑ f (x₁, (v_{i₁})₂, x₃, ...), where at each index j one uses either xⱼ or one of the (vᵢ)ⱼ, and each vᵢ has to be used exactly once. The sum is parameterized by the embeddings of Fin k in the index type ι (or, equivalently, by the subsets s of ι of cardinality k and then the bijections between Fin k and s).

The fact that this is indeed the iterated Fréchet derivative is proved in ContinuousMultilinearMap.iteratedFDeriv_eq.

Equations

• f.iteratedFDeriv k x = ∑ e : Fin k ↪ ι, (f.iteratedFDerivComponent e.toEquivRange) ((Pi.compRightL 𝕜 E₁ Subtype.val) x)

theorem ContinuousMultilinearMap.norm_iteratedFDeriv_le' {𝕜 : Type u} {ι : Type v} {E₁ : ι → Type wE₁} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E₁ i)] [(i : ι) → NormedSpace 𝕜 (E₁ i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E₁ G) (k : ℕ) (x : (i : ι) → E₁ i) : ‖f.iteratedFDeriv k x‖ ≤ ↑((Fintype.card ι).descFactorial k) * ‖f‖ * ‖x‖ ^ (Fintype.card ι - k)

Controlling the norm of f.iteratedFDeriv when f is continuous multilinear. For the same bound on the iterated derivative of f in the calculus sense, see ContinuousMultilinearMap.norm_iteratedFDeriv_le.

Results that are only true if the target space is a NormedAddCommGroup (and not just a SeminormedAddCommGroup).

theorem ContinuousMultilinearMap.opNorm_zero_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ‖f‖ = 0 ↔ f = 0

A continuous linear map is zero iff its norm vanishes.

@[deprecated ContinuousMultilinearMap.opNorm_zero_iff]

theorem ContinuousMultilinearMap.op_norm_zero_iff {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] (f : ContinuousMultilinearMap 𝕜 E G) : ‖f‖ = 0 ↔ f = 0

Alias of ContinuousMultilinearMap.opNorm_zero_iff.

---

A continuous linear map is zero iff its norm vanishes.

noncomputable instance ContinuousMultilinearMap.normedAddCommGroup {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] : NormedAddCommGroup (ContinuousMultilinearMap 𝕜 E G)

Continuous multilinear maps themselves form a normed group with respect to the operator norm.

Equations

• ContinuousMultilinearMap.normedAddCommGroup = NormedAddCommGroup.ofSeparation ⋯

noncomputable instance ContinuousMultilinearMap.normedAddCommGroup' {𝕜 : Type u} {ι : Type v} {G : Type wG} {G' : Type wG'} [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [SeminormedAddCommGroup G'] [NormedSpace 𝕜 G'] : NormedAddCommGroup (ContinuousMultilinearMap 𝕜 (fun (x : ι) => G') G)

An alias of ContinuousMultilinearMap.normedAddCommGroup with non-dependent types to help typeclass search.

Equations

• ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup

theorem ContinuousMultilinearMap.norm_ofSubsingleton_id (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] [Nontrivial G] (i : ι) : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖ = 1

theorem ContinuousMultilinearMap.nnnorm_ofSubsingleton_id (𝕜 : Type u) {ι : Type v} (G : Type wG) [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [Subsingleton ι] [Nontrivial G] (i : ι) : ‖(ContinuousMultilinearMap.ofSubsingleton 𝕜 G G i) (ContinuousLinearMap.id 𝕜 G)‖₊ = 1

noncomputable instance ContinuousMultilinearMap.completeSpace {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → SeminormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [NormedAddCommGroup G] [NormedSpace 𝕜 G] [CompleteSpace G] : CompleteSpace (ContinuousMultilinearMap 𝕜 E G)

If the target space is complete, the space of continuous multilinear maps with its norm is also complete. The proof is essentially the same as for the space of continuous linear maps (modulo the addition of Finset.prod where needed). The duplication could be avoided by deducing the linear case from the multilinear case via a currying isomorphism. However, this would mess up imports, and it is more satisfactory to have the simplest case as a standalone proof.

Equations

• ⋯ = ⋯

Results that are only true if the source is a NormedAddCommGroup (and not just a SeminormedAddCommGroup).

theorem MultilinearMap.bound_of_shell {𝕜 : Type u} {ι : Type v} {E : ι → Type wE} {G : Type wG} [Fintype ι] [NontriviallyNormedField 𝕜] [(i : ι) → NormedAddCommGroup (E i)] [(i : ι) → NormedSpace 𝕜 (E i)] [SeminormedAddCommGroup G] [NormedSpace 𝕜 G] (f : MultilinearMap 𝕜 E G) {ε : ι → ℝ} {C : ℝ} (hε : ∀ (i : ι), 0 < ε i) {c : ι → 𝕜} (hc : ∀ (i : ι), 1 < ‖c i‖) (hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖) (m : (i : ι) → E i) : ‖f m‖ ≤ C * ∏ i : ι, ‖m i‖

If a multilinear map in finitely many variables on normed spaces satisfies the inequality ‖f m‖ ≤ C * ∏ i, ‖m i‖ on a shell ε i / ‖c i‖ < ‖m i‖ < ε i for some positive numbers ε i and elements c i : 𝕜, 1 < ‖c i‖, then it satisfies this inequality for all m.