Proposition 4

Tag 05RX, permanent. Cite it as [Bourbaki, Tag 05RX].

Every subspace $F$ of a vector space $E$ is a direct factor of $E$ and

\dim F + \dim(E/F) = \dim E.

As the quotient vector space $E/F$ is a free module, we know (§ 1, no. 11, Proposition 21) that $F$ is a direct factor of $E$ ; relation (3) is then a special case of formula (1) (no. 2).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Remark, § 7
Remark, § 7
Proposition 3, § 7
Theorem 4, § 7
Theorem 5, § 7
Proposition 6, § 8
Remark, § 5

Cites

§ 1, no. 11, Proposition 21