---
tag: WJ7PDMGQQW
kind: concept
name: Hilbert sum
msc: [46C05]
sources:
  - url: https://en.wikipedia.org/wiki/Direct_sum
---

## Summary

The Hilbert sum of a family of Hilbert spaces consists of the families of vectors, one from each space, with finite sum of squared norms.
It is the right notion of direct sum for Hilbert spaces, because the algebraic direct sum of infinitely many of them is not complete.
Every Hilbert space with an orthonormal basis indexed by $I$ is the Hilbert sum of $I$ copies of the scalar field.

## Explanations

### L1 human

Think of putting several Hilbert spaces side by side at right angles to each other.
A vector of the sum has one component in each space, and its squared length is the sum of the squared lengths of the components.
With infinitely many spaces, one only keeps families for which this sum is finite.
For example, the Hilbert sum of countably many copies of $\R$ is the space $\ell^2$ of square-summable real sequences.