Theorems · Theorem · functional analysis

LinearMap.finrank_range_adjoint

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedAddCommGroup F] [inst_3 : InnerProductSpace 𝕜 E] [inst_4 : InnerProductSpace 𝕜 F]
  [inst_5 : FiniteDimensional 𝕜 E] [inst_6 : FiniteDimensional 𝕜 F] (A : E →ₗ[𝕜] F),
  Module.finrank 𝕜 ↥(LinearMap.adjoint A).range = Module.finrank 𝕜 ↥A.range

Part of 7.64(d) in [axler2024].

Defined in

Mathlib.Analysis.InnerProductSpace.Adjoint

Cited by

1 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedAddCommGroupInnerProductSpaceInnerProductSpaceFiniteDimensionalFiniteDimensional