(Analysis Now, Pederson) Let $H_\alpha$ be a collection of Hilbert spaces. Let $\sum_\alpha H_\alpha $ be the algebraic direct sum of the $H_\alpha$, i.e. the space of $x \in \prod_\alpha H_\alpha $ such that $x_\alpha = 0$ except finitely many $\alpha$. Put the inner product $\left< x, y \right> = \sum_\alpha \left< x_\alpha, y_\alpha \right>$ on the algebraic direct sum. Call the completion of the algebraic direct sum $\bigoplus_\alpha H_\alpha$ the orthogonal sum of the $H_\alpha$.
"We may identify each $H_\alpha$ with a closed subspace of the orthogonal sum such that $H_\alpha \perp H_\beta$ for $\alpha \neq \beta$." What is this identification, specifically?
I see that $H_\alpha^\prime := \sum_\beta H_\beta^\prime$ is a subspace of the algebraic direct sum, where $H_\beta^\prime$ equals $H_\alpha$ if $\beta=\alpha$ and is $0$ otherwise, and $H_\alpha^\prime \perp H_\gamma^\prime$. I am thinking then to take the corresponding completion isometry $J: \sum_\beta H_\beta \rightarrow \bigoplus_\beta H_\beta$ and consider the $J(H_\alpha^\prime)$, but I'm not so sure.