Proposition 2

— Let E be a hilbertian space and $(V_\lambda)_{\lambda \in L}$ a family of closed vector subspaces of E; for every $\lambda \in L$, let $(W_{\lambda \mu})_{\mu \in M_\lambda}$ be a family of closed vector subspaces of $V_\lambda$ such that $V_\lambda$ is the closed vector subspace generated by the union of this family. In order that E is the hilbertian sum of the family $(W_{\lambda \mu})_{\lambda \in L, \mu \in M_\lambda}$, it is necessary and sufficient that E is the hilbertian sum of the family $(V_\lambda)_{\lambda \in L}$ and that, for each $\lambda \in L$, $V_\lambda$ is the hilbertian sum of the family $(W_{\lambda \mu})_{\mu \in M_\lambda}$ ("associativity of the hilbertian sum").

To show that the condition is necessary, it is enough to see that $V_\alpha$ and $V_\beta$ are orthogonal if $\alpha \neq \beta$. But, every element of $W_{\alpha \mu}$ ($\mu \in M_\alpha$) is orthogonal to all the $W_{\beta v}$ ($v \in M_\beta$), hence to the closed vector subspace $V_\beta$ which they generate; the same argument then shows that every element of $V_\beta$ is orthogonal to $V_\alpha$, being orthogonal to all the $W_{\alpha \mu}$ ($\mu \in M_\alpha$).

To show that the condition is sufficient, it is enough to verify that, if it is satisfied, E is equal to the closed vector subspace F generated by the union of the $W_{\lambda \mu}$ ($\lambda \in L$, μ ∈ M_λ); but, for each λ ∈ L, F contains the closed vector subspace generated by the union of the W_{λμ} such that μ ∈ M_λ, that is, F contains V_λ; hence F is the closed vector subspace generated by the union of the V_λ, which is E by hypothesis.