Theorem 2
Tag 05RJ, permanent. Cite it as [Bourbaki, Tag 05RJ].
Algebra, chapter II, § 7. Vector spaces
Given a generating system S of a vector space E over a field K and a free subset L of E contained in S, there exists a basis B of E such that L ⊂ B ⊂ S.
Theorem 1 follows from this statement by taking L = ∅.
To prove Theorem 2, we note that the set Λ of free subsets of E contained in S, ordered by inclusion, is an inductive set (Set Theory, III, § 2, no. 4), by virtue of § 1, no. 11; so is the set M of free subsets containing L and contained in S. By Zorn's Lemma, M admits a maximal element B and it suffices to prove that the vector subspace of E generated by B is equal to E. This follows immediately from the definition of B and the following lemma:
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
- Corollary 1, § 3
- Proposition 13, § 10
- Corollary, § 7
- Corollary, § 7
- Proposition 19, § 7
- Corollary 3, § 7
- Theorem 3, § 7
- Corollary 3, § 7
- Lemma 1, § 8
- Proposition 1, § 8
- Proposition 7, § 4