Theorems · Theorem · functional analysis

LinearMap.normDet_eq_zero_iff_rank_range_ne

∀ {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 ↔ Module.finrank 𝕜 ↥f.range ≠ Module.finrank 𝕜 U

Defined in

Mathlib.Analysis.InnerProductSpace.NormDet

Cited by

3 results in Mathlib

Foundations

Depth 245 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceFiniteDimensionalNormedAddCommGroupInnerProductSpace