Theorems · Theorem · functional analysis

GeneralSchauderBasis.finrank_range_proj

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {X : Type u_2} [inst_1 : NormedAddCommGroup X]
  [inst_2 : NormedSpace 𝕜 X] {β : Type u_3} {L : SummationFilter β} (b : GeneralSchauderBasis β 𝕜 X L) (A : Finset β),
  Module.finrank 𝕜 ↥(↑(b.proj A)).range = A.card

The dimension of the range of the projection proj A equals the cardinality of A.

Defined in: Mathlib.Analysis.Normed.Module.Bases
Cited by: 1 results in Mathlib
Foundations: Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField NormedAddCommGroup NormedSpace

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

Setproof · cited by 53,352
RingHom.idstatement · cited by 18,349
NormedAddCommGroupstatement and proof · cited by 15,752
Finsetstatement and proof · cited by 13,712
NormedSpacestatement and proof · cited by 12,499
NontriviallyNormedFieldstatement and proof · cited by 8,742
SetLike.coeproof · cited by 8,199
Submodulestatement and proof · cited by 7,192
Finset.cardstatement and proof · cited by 2,327
Module.finrankstatement and proof · cited by 1,770
Submodule.spanproof · cited by 1,504
LinearMap.rangestatement · cited by 893

Cited by1

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SchauderBasis.finrank_range_projproof · cited by 0