Theorems · Theorem · functional analysis

IsBoundedBilinearMap.bound

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [inst : Semiring 𝕜] [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : Module 𝕜 E] [inst_3 : SeminormedAddCommGroup F] [inst_4 : Module 𝕜 F] [inst_5 : SeminormedAddCommGroup G]
  [inst_6 : Module 𝕜 G] {f : E × F → G},
  IsBoundedBilinearMap 𝕜 f → ∃ C > 0, ∀ (x : E) (y : F), ‖f (x, y)‖ ≤ C * ‖x‖ * ‖y‖

Defined in: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
Cited by: 3 results in Mathlib
Foundations: Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring SeminormedAddCommGroup Module SeminormedAddCommGroup Module SeminormedAddCommGroup Module

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Cites6

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement · cited by 25,697
Modulestatement and proof · cited by 20,661
Semiringstatement and proof · cited by 13,802
Norm.normstatement · cited by 5,413
SeminormedAddCommGroupstatement and proof · cited by 2,671
IsBoundedBilinearMapstatement and proof · cited by 40

Cited by3

Results whose statement or proof uses this declaration.

IsBoundedBilinearMap.isBoundedLinearMap_rightproof · cited by 3
IsBoundedBilinearMap.symmproof · cited by 2
IsBoundedBilinearMap.isBigOproof · cited by 2