$E$ is a vectorial space equipped with an inner product $\langle \cdot, \cdot \rangle$. $(E_i)_{i \in I}$ is a family of complete pairwise orthogonal subspaces.
Is the subspace $V$ generated by the union of the $E_i$ closed?
I ask the question as I'm reading a French book on Topology and functional analysis. And in a theorem it is mentionned $V$ is the closed subspace generated par the union of the $E_i$. So I have issues to understand if $V$ is closed as a consequence of the hypothesis or if it is an additional hypothesis.
And I'm not to able to prove that $V$ is closed based on the other hypothesis...