---
tag: DE42T92C4R
kind: definition
name: Hilbert sum
msc: [46C05]
defines: WJ7PDMGQQW
uses: [HM2HHE3J51, 0S7W6EG32V, 0XJXTBCB2T]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_sum
---

## Definition

Let $\mathbb{K}$ be $\R$ or $\C$, and let $(H_i)_{i \in I}$ be a family of [[Hilbert spaces|HM2HHE3J51]] over $\mathbb{K}$.
The *Hilbert sum* $\widehat{\bigoplus}_{i \in I} H_i$ is the set of families $x = (x_i)_{i \in I}$ with $x_i \in H_i$ for each $i$ and $\sum_{i \in I} \norm{x_i}^2 < \infty$.
Its vector operations are componentwise, and its inner product is
$$\ip{x}{y} = \sum_{i \in I} \ip{x_i}{y_i}.$$
With this inner product it is a Hilbert space.

A Hilbert space $H$ is the *internal Hilbert sum* of a family $(V_i)_{i \in I}$ of closed subspaces when these two conditions hold:

1. $V_i \perp V_j$ for all $i \neq j$.
2. The linear span of $\bigcup_{i \in I} V_i$ is dense in $H$.

## 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|0S7W6EG32V]] $\bigoplus_{i} H_i$ 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|0XJXTBCB2T]] $V^\perp$.