Theorems · Theorem · functional analysis

InnerProductSpace.span_gramSchmidt

∀ (𝕜 : Type u_1) {E : Type u_2} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup E] [inst_2 : InnerProductSpace 𝕜 E]
  {ι : Type u_3} [inst_3 : LinearOrder ι] [inst_4 : LocallyFiniteOrderBot ι] [inst_5 : WellFoundedLT ι] (f : ι → E),
  Submodule.span 𝕜 (Set.range (InnerProductSpace.gramSchmidt 𝕜 f)) = Submodule.span 𝕜 (Set.range f)

gramSchmidt preserves span of vectors.

Defined in: Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
Cited by: 0 results in Mathlib
Foundations: Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: RCLike NormedAddCommGroup InnerProductSpace LinearOrder LocallyFiniteOrderBot WellFoundedLT

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NormedAddCommGroupstatement and proof · cited by 15,752
LinearOrderstatement and proof · cited by 8,572
Submodulestatement · cited by 7,192
Set.rangestatement · cited by 4,705
InnerProductSpacestatement and proof · cited by 3,523
RCLikestatement and proof · cited by 2,829
le_rflproof · cited by 1,558
Submodule.spanstatement · cited by 1,504
Set.Iicproof · cited by 1,111
WellFoundedLTstatement and proof · cited by 491
LocallyFiniteOrderBotstatement and proof · cited by 286
Set.range_subset_iffproof · cited by 99

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