Corollary 4

Tag 05S1, permanent. Cite it as [Bourbaki, Tag 05S1].

For every subspace F of a vector space E, $\dim F \leq \dim E$ ; if E is finite dimensional, the relation $\dim F = \dim E$ is equivalent to $F = E$ .

The first assertion is obvious from (3); further, if $\dim E$ is finite, the relation $\dim F = \dim E$ implies $\dim(E/F) = 0$ by (3) and a vector space of dimension 0 reduces to 0.

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

Cited by

Proposition 12, § 10
Remark, § 7
Remark, § 7
Theorem 4, § 7

Cites

Nothing.