Hilbert sum

Remarks

The series defining $\ip{x}{y}$ converges absolutely, because $\abs{\ip{x_i}{y_i}} \le \tfrac12 (\norm{x_i}^2 + \norm{y_i}^2)$. If $H$ is the internal Hilbert sum of $(V_i)$, then $x \mapsto (P_i x)_{i}$, where $P_i$ is the orthogonal projection onto $V_i$, is a unitary map from $H$ onto $\widehat{\bigoplus} V_i$.

Example: if every $H_i$ is $\mathbb{K}$, the Hilbert sum is $\ell^2(I)$. For each $i$, let $e_i$ be the family with $1$ at $i$ and $0$ elsewhere; the lines $\mathbb{K} e_i$ are pairwise orthogonal and their span, the finitely supported families, is dense.

Non-example: for infinite $I$ and all $H_i \neq 0$, the external direct sum $\bigoplus_{i} H_i$ of finitely supported families is not complete. So it is not a Hilbert space for the same inner product. The Hilbert sum is its completion.

For a closed subspace $V$ of $H$, the space $H$ is the internal Hilbert sum of $V$ and its orthogonal complement $V^\perp$.