Product of normed groups and other constructions #

theorem Pi.seminormedAddGroup.proof_1 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) :

↑(Finset.univ.sup fun (b : ι) => nndist (x b) (y b)) = ↑(Finset.univ.sup fun (b : ι) => ‖(x - y) b‖₊)

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instance Pi.seminormedGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] :

SeminormedGroup ((i : ι) → π i)

Finite product of seminormed groups, using the sup norm.

Equations

Pi.seminormedGroup = SeminormedGroup.mk ⋯

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theorem Pi.norm_def {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) :

‖f‖ = ↑(Finset.univ.sup fun (b : ι) => ‖f b‖₊)

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theorem Pi.norm_def' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) :

‖f‖ = ↑(Finset.univ.sup fun (b : ι) => ‖f b‖₊)

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theorem Pi.nnnorm_def {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) :

‖f‖₊ = Finset.univ.sup fun (b : ι) => ‖f b‖₊

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theorem Pi.nnnorm_def' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) :

‖f‖₊ = Finset.univ.sup fun (b : ι) => ‖f b‖₊

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theorem pi_norm_le_iff_of_nonneg {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 ≤ r) :

‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r

The seminorm of an element in a product space is ≤ r if and only if the norm of each component is.

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theorem pi_norm_le_iff_of_nonneg' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 ≤ r) :

‖x‖ ≤ r ↔ ∀ (i : ι), ‖x i‖ ≤ r

The seminorm of an element in a product space is ≤ r if and only if the norm of each component is.

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theorem pi_nnnorm_le_iff {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : NNReal} :

‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r

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theorem pi_nnnorm_le_iff' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : NNReal} :

‖x‖₊ ≤ r ↔ ∀ (i : ι), ‖x i‖₊ ≤ r

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theorem pi_norm_le_iff_of_nonempty {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) {r : ℝ} [Nonempty ι] :

‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r

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theorem pi_norm_le_iff_of_nonempty' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) {r : ℝ} [Nonempty ι] :

‖f‖ ≤ r ↔ ∀ (b : ι), ‖f b‖ ≤ r

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theorem pi_norm_lt_iff {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 < r) :

‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r

The seminorm of an element in a product space is < r if and only if the norm of each component is.

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theorem pi_norm_lt_iff' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : ℝ} (hr : 0 < r) :

‖x‖ < r ↔ ∀ (i : ι), ‖x i‖ < r

The seminorm of an element in a product space is < r if and only if the norm of each component is.

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theorem pi_nnnorm_lt_iff {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] {x : (i : ι) → π i} {r : NNReal} (hr : 0 < r) :

‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r

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theorem pi_nnnorm_lt_iff' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] {x : (i : ι) → π i} {r : NNReal} (hr : 0 < r) :

‖x‖₊ < r ↔ ∀ (i : ι), ‖x i‖₊ < r

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theorem norm_le_pi_norm {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) (i : ι) :

‖f i‖ ≤ ‖f‖

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theorem norm_le_pi_norm' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) (i : ι) :

‖f i‖ ≤ ‖f‖

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theorem nnnorm_le_pi_nnnorm {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) (i : ι) :

‖f i‖₊ ≤ ‖f‖₊

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theorem nnnorm_le_pi_nnnorm' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) (i : ι) :

‖f i‖₊ ≤ ‖f‖₊

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theorem pi_norm_const_le {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] (a : E) :

‖fun (x : ι) => a‖ ≤ ‖a‖

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theorem pi_norm_const_le' {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] (a : E) :

‖fun (x : ι) => a‖ ≤ ‖a‖

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theorem pi_nnnorm_const_le {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] (a : E) :

‖fun (x : ι) => a‖₊ ≤ ‖a‖₊

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theorem pi_nnnorm_const_le' {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] (a : E) :

‖fun (x : ι) => a‖₊ ≤ ‖a‖₊

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@[simp]

theorem pi_norm_const {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] [Nonempty ι] (a : E) :

‖fun (_i : ι) => a‖ = ‖a‖

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@[simp]

theorem pi_norm_const' {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] [Nonempty ι] (a : E) :

‖fun (_i : ι) => a‖ = ‖a‖

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@[simp]

theorem pi_nnnorm_const {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedAddGroup E] [Nonempty ι] (a : E) :

‖fun (_i : ι) => a‖₊ = ‖a‖₊

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@[simp]

theorem pi_nnnorm_const' {ι : Type u_1} {E : Type u_2} [Fintype ι] [SeminormedGroup E] [Nonempty ι] (a : E) :

‖fun (_i : ι) => a‖₊ = ‖a‖₊

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theorem Pi.sum_norm_apply_le_norm {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) :

∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖

The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality.

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theorem Pi.sum_norm_apply_le_norm' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) :

∑ i : ι, ‖f i‖ ≤ Fintype.card ι • ‖f‖

The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality.

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theorem Pi.sum_nnnorm_apply_le_nnnorm {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddGroup (π i)] (f : (i : ι) → π i) :

∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊

The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality.

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theorem Pi.sum_nnnorm_apply_le_nnnorm' {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedGroup (π i)] (f : (i : ι) → π i) :

∑ i : ι, ‖f i‖₊ ≤ Fintype.card ι • ‖f‖₊

The $L^1$ norm is less than the $L^\infty$ norm scaled by the cardinality.

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instance Pi.seminormedAddCommGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (π i)] :

SeminormedAddCommGroup ((i : ι) → π i)

Finite product of seminormed groups, using the sup norm.

Equations

Pi.seminormedAddCommGroup = let __src := Pi.seminormedAddGroup; SeminormedAddCommGroup.mk ⋯

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theorem Pi.seminormedAddCommGroup.proof_1 {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → SeminormedAddCommGroup (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :

a + b = b + a

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theorem Pi.seminormedAddCommGroup.proof_2 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → SeminormedAddCommGroup (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) :

dist x y = ‖x - y‖

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instance Pi.seminormedCommGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → SeminormedCommGroup (π i)] :

SeminormedCommGroup ((i : ι) → π i)

Finite product of seminormed groups, using the sup norm.

Equations

Pi.seminormedCommGroup = let __src := Pi.seminormedGroup; SeminormedCommGroup.mk ⋯

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theorem Pi.normedAddGroup.proof_2 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → NormedAddGroup (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) :

dist x y = ‖x - y‖

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theorem Pi.normedAddGroup.proof_1 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → NormedAddGroup (π i)] :

∀ {x y : (i : ι) → π i}, dist x y = 0 → x = y

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instance Pi.normedAddGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → NormedAddGroup (π i)] :

NormedAddGroup ((i : ι) → π i)

Finite product of seminormed groups, using the sup norm.

Equations

Pi.normedAddGroup = let __src := Pi.seminormedAddGroup; NormedAddGroup.mk ⋯

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instance Pi.normedGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → NormedGroup (π i)] :

NormedGroup ((i : ι) → π i)

Finite product of normed groups, using the sup norm.

Equations

Pi.normedGroup = let __src := Pi.seminormedGroup; NormedGroup.mk ⋯

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theorem Pi.normedAddCommGroup.proof_2 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → NormedAddCommGroup (π i)] :

∀ {x y : (i : ι) → π i}, dist x y = 0 → x = y

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theorem Pi.normedAddCommGroup.proof_3 {ι : Type u_1} {π : ι → Type u_2} [Fintype ι] [(i : ι) → NormedAddCommGroup (π i)] (x : (i : ι) → π i) (y : (i : ι) → π i) :

dist x y = ‖x - y‖

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theorem Pi.normedAddCommGroup.proof_1 {ι : Type u_1} {π : ι → Type u_2} [(i : ι) → NormedAddCommGroup (π i)] (a : (i : ι) → π i) (b : (i : ι) → π i) :

a + b = b + a

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instance Pi.normedAddCommGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → NormedAddCommGroup (π i)] :

NormedAddCommGroup ((i : ι) → π i)

Finite product of seminormed groups, using the sup norm.

Equations

Pi.normedAddCommGroup = let __src := Pi.seminormedAddGroup; NormedAddCommGroup.mk ⋯

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instance Pi.normedCommGroup {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [(i : ι) → NormedCommGroup (π i)] :

NormedCommGroup ((i : ι) → π i)

Finite product of normed groups, using the sup norm.

Equations

Pi.normedCommGroup = let __src := Pi.seminormedGroup; NormedCommGroup.mk ⋯

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theorem Pi.nnnorm_single {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [DecidableEq ι] [(i : ι) → NormedAddCommGroup (π i)] {i : ι} (y : π i) :

‖Pi.single i y‖₊ = ‖y‖₊

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theorem Pi.norm_single {ι : Type u_1} {π : ι → Type u_4} [Fintype ι] [DecidableEq ι] [(i : ι) → NormedAddCommGroup (π i)] {i : ι} (y : π i) :

‖Pi.single i y‖ = ‖y‖

Multiplicative opposite #

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instance MulOpposite.instSeminormedAddGroup {E : Type u_2} [SeminormedAddGroup E] :

SeminormedAddGroup Eᵐᵒᵖ

The (additive) norm on the multiplicative opposite is the same as the norm on the original type.

Note that we do not provide this more generally as Norm Eᵐᵒᵖ, as this is not always a good choice of norm in the multiplicative SeminormedGroup E case.

We could repeat this instance to provide a [SeminormedGroup E] : SeminormedGroup Eᵃᵒᵖ instance, but that case would likely never be used.

Equations

MulOpposite.instSeminormedAddGroup = let __spread.0 := MulOpposite.instPseudoMetricSpace; SeminormedAddGroup.mk ⋯

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theorem MulOpposite.norm_op {E : Type u_2} [SeminormedAddGroup E] (a : E) :

‖MulOpposite.op a‖ = ‖a‖

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theorem MulOpposite.norm_unop {E : Type u_2} [SeminormedAddGroup E] (a : Eᵐᵒᵖ) :

‖MulOpposite.unop a‖ = ‖a‖

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theorem MulOpposite.nnnorm_op {E : Type u_2} [SeminormedAddGroup E] (a : E) :

‖MulOpposite.op a‖₊ = ‖a‖₊

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theorem MulOpposite.nnnorm_unop {E : Type u_2} [SeminormedAddGroup E] (a : Eᵐᵒᵖ) :

‖MulOpposite.unop a‖₊ = ‖a‖₊

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instance MulOpposite.instNormedAddGroup {E : Type u_2} [NormedAddGroup E] :

NormedAddGroup Eᵐᵒᵖ

Equations

MulOpposite.instNormedAddGroup = let __spread.0 := MulOpposite.instMetricSpace; let __spread.1 := MulOpposite.instSeminormedAddGroup; NormedAddGroup.mk ⋯

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instance MulOpposite.instSeminormedAddCommGroup {E : Type u_2} [SeminormedAddCommGroup E] :

SeminormedAddCommGroup Eᵐᵒᵖ

Equations

MulOpposite.instSeminormedAddCommGroup = SeminormedAddCommGroup.mk ⋯

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instance MulOpposite.instNormedAddCommGroup {E : Type u_2} [NormedAddCommGroup E] :

NormedAddCommGroup Eᵐᵒᵖ

Equations

MulOpposite.instNormedAddCommGroup = let __spread.0 := MulOpposite.instSeminormedAddCommGroup; let __spread.1 := MulOpposite.instNormedAddGroup; NormedAddCommGroup.mk ⋯

Documentation

Mathlib.Analysis.Normed.Group.Constructions

Product of normed groups and other constructions #

`PUnit` #

`ULift` #

`Additive`, `Multiplicative` #

Order dual #

Binary product of normed groups #

Finite product of normed groups #

Multiplicative opposite #

Product of normed groups and other constructions #

PUnit #

ULift #

Additive, Multiplicative #

Order dual #

Binary product of normed groups #

Finite product of normed groups #

Multiplicative opposite #

`PUnit` #

`ULift` #

`Additive`, `Multiplicative` #