Remark 2

Let M be a closed vector subspace of E and $(\dot{a}_i)_{i \in I}$ a topologically independent family in the quotient space E/M. If $a_i$ is any element of the class $\dot{a}_i$, then from def. 2, and the fact that the canonical mapping of E on E/M is continuous, it follows that the family $(a_i)_{i \in I}$ is topologically independent. But note that if N is the closed vector subspace generated by the $a_i$ it can happen that $M \cap N \neq \{0\}$ (I, p. 25, exerc. 2), and hence the sum $M + N$ is not necessarily direct in the algebraic sense (nor a fortiori in the topological sense).