Hilbert 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.