Theorems · Definition · linear algebra

Module.Basis.ofVectorSpace

(K : Type u_3) →
  (V : Type u_4) →
    [inst : DivisionRing K] →
      [inst_1 : AddCommGroup V] → [inst_2 : Module K V] → Module.Basis (↑(Module.Basis.ofVectorSpaceIndex K V)) K V

Each vector space has a basis.

Defined in: Mathlib.LinearAlgebra.Basis.VectorSpace
Cited by: 30 results in Mathlib
Foundations: Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: DivisionRing AddCommGroup Module

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Cites7

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

Modulestatement and proof · cited by 20,661
AddCommGroupstatement and proof · cited by 12,871
Set.Elemstatement · cited by 7,166
Module.Basisstatement · cited by 1,477
DivisionRingstatement and proof · cited by 1,062
Module.Basis.ofVectorSpaceIndexstatement · cited by 29
Module.Basis.extendproof · cited by 9

Cited by33

Results whose statement or proof uses this declaration.

Submodule.eq_top_of_finrank_eqproof · cited by 15
LinearMap.exists_leftInverse_of_injectiveproof · cited by 6
Subspace.dual_finrank_eqproof · cited by 5
IsNoetherian.finsetBasisproof · cited by 5
Module.Basis.coe_ofVectorSpacestatement · cited by 4
LocallyBoundedVariationOn.ae_differentiableWithinAt_of_memproof · cited by 4
coevaluationproof · cited by 4
Module.Basis.ofVectorSpace_apply_selfstatement · cited by 3
coevaluation_apply_onestatement and proof · cited by 3
Module.Basis.exists_basisproof · cited by 2
lipschitzExtensionConstant_defstatement and proof · cited by 2
Module.Basis.range_ofVectorSpacestatement · cited by 2