Theorems · Theorem · functional analysis

IsBoundedLinearMap.smul

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : Semiring 𝕜] [inst_1 : SeminormedAddCommGroup E]
  [inst_2 : Module 𝕜 E] [inst_3 : SeminormedAddCommGroup F] [inst_4 : Module 𝕜 F] {f : E → F} {𝕜' : Type u_5} (c : 𝕜')
  [inst_5 : SeminormedRing 𝕜'] [inst_6 : Module 𝕜' F] [IsBoundedSMul 𝕜' F] [SMulCommClass 𝕜 𝕜' F],
  IsBoundedLinearMap 𝕜 f → IsBoundedLinearMap 𝕜 (c • f)

Defined in: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
Cited by: 0 results in Mathlib
Foundations: Depth 114 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Semiring SeminormedAddCommGroup Module SeminormedAddCommGroup Module SeminormedRing Module IsBoundedSMul SMulCommClass

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Realproof · cited by 25,697
Modulestatement and proof · cited by 20,661
Semiringstatement and proof · cited by 13,802
Norm.normproof · cited by 5,413
SeminormedAddCommGroupstatement and proof · cited by 2,671
le_reflproof · cited by 2,061
SMulCommClassstatement and proof · cited by 1,927
mul_assocproof · cited by 1,667
norm_nonnegproof · cited by 725
le_imp_le_of_le_of_leproof · cited by 576
SeminormedRingstatement and proof · cited by 446
mul_le_mul_of_nonneg_leftproof · cited by 361

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