Theorems · Theorem · functional analysis

LinearMap.normDet_eq_zero_tfae

∀ {𝕜 : Type u_1} {U : Type u_2} {V : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup U]
  [inst_2 : InnerProductSpace 𝕜 U] [inst_3 : FiniteDimensional 𝕜 U] [inst_4 : NormedAddCommGroup V]
  [inst_5 : InnerProductSpace 𝕜 V] (f : U →ₗ[𝕜] V),
  [f.normDet = 0, f.ker ≠ ⊥, Module.finrank 𝕜 ↥f.range ≠ Module.finrank 𝕜 U,
      Module.finrank 𝕜 ↥f.range < Module.finrank 𝕜 U, IsEmpty (OrthonormalBasis (Fin (Module.finrank 𝕜 U)) 𝕜 ↥f.range),
      ¬Function.Injective ⇑f].TFAE

Defined in

Mathlib.Analysis.InnerProductSpace.NormDet

Cited by

3 results in Mathlib

Foundations

Depth 247 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpace