Proposition 2

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

Let (Et)_{t \in I} be a family of vector spaces over K. Then*

\dim_K \left( \bigoplus_{t \in I} E_t \right) = \sum_{t \in I} \dim_K E_t.

If the E_t are canonically identified with subspaces of $E = \bigoplus_{t \in I} E_t$ and B_t is a basis of E_t ( $t \in I$ ), then $B = \bigcup_{t \in I} B_t$ is a basis of E ( $§ 1$ , no. 11, Proposition 19); whence relation (1) since the B_t are pairwise disjoint.

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

Cited by

Remark, § 7
Corollary 3, § 7
Proposition 6, § 7

Cites

no. 11, Proposition 19