Theorems · Theorem · functional analysis

ContinuousLinearMap.isOpen_injective

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type v} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type w} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] [CompleteSpace 𝕜]
  [FiniteDimensional 𝕜 E], IsOpen {L | Function.Injective ⇑L}

The set of injective continuous linear maps E → F is open, if E is finite-dimensional over a complete field.

Defined in

Mathlib.Analysis.Normed.Module.FiniteDimension

Cited by

0 results in Mathlib

Foundations

Depth 178 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceFiniteDimensional