Corollary 5

If a vector space E is the sum of a family $(F_i)$ of vector subspaces, then

(8)

If further $\dim E$ is finite, the two sides of (8) are equal if and only if E is the direct sum of the family $(F_i)$.

The inequality (8) follows from (3) and the fact that E is isomorphic to a quotient of $\bigoplus_i F_i$ (§ 1, no. 7, formula (28)). The second assertion is a particular case of § 1, no. 10, Corollary 5 to Proposition 16, for the equality of the two sides of (8) implies that $\dim F_i = 0$ except for a finite number of indices.