Theorems · Theorem · functional analysis
isBoundedLinearMap_iff
∀ {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},
IsBoundedLinearMap 𝕜 f ↔ IsLinearMap 𝕜 f ∧ ∃ M, 0 < M ∧ ∀ (x : E), ‖f x‖ ≤ M * ‖x‖- Cited by
- 0 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites9
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- Norm.normstatement and proof · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- IsLinearMapstatement and proof · cited by 41
- IsBoundedLinearMapstatement and proof · cited by 39
- IsBoundedLinearMap.toIsLinearMapproof · cited by 7
- IsBoundedLinearMap.boundproof · cited by 4
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