Theorems · Theorem · functional analysis

ContinuousLinearEquiv.isOpen

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : SeminormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F]
  [CompleteSpace E], IsOpen (Set.range ContinuousLinearEquiv.toContinuousLinearMap)

Defined in: Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
Cited by: 1 results in Mathlib
Foundations: Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField NormedAddCommGroup NormedSpace SeminormedAddCommGroup NormedSpace CompleteSpace

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Cites36

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

DFunLike.coeproof · cited by 62,936
Setproof · cited by 53,352
TopologicalSpaceproof · cited by 24,529
RingHom.idstatement and proof · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
NontriviallyNormedFieldstatement and proof · cited by 8,742
Set.ofPredproof · cited by 6,101
ContinuousLinearMapstatement and proof · cited by 5,352
Set.preimageproof · cited by 4,946
Set.rangestatement and proof · cited by 4,705
Unitsproof · cited by 2,804

Cited by1

Results whose statement or proof uses this declaration.

ContinuousLinearEquiv.nhdsproof · cited by 2