direct sum of Hilbert spaces

Let ${\{H_i\}}_{i\in I}$ be a family of Hilbert spaces indexed by a set $I$. The direct sum of this family of Hilbert spaces, denoted as

$$\underset{i\in I}{\oplus }{H}_i$$

consists of all elements $v$ of the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) of ${\{H_i\}}_{i\in I}$ such that . Of course, for the previous sum to be finite only at most a countable number of $v_i$ can be non-zero.

Vector addition and scalar multiplication are defined termwise: If $u,v\in {\oplus }_{i\in I}{H}_i$, then ${(u+v)}_i=u_i+v_i$ and ${(sv)}_i=s{v}_i$.

The inner product of two vectors is defined as

$$⟨u,v⟩=\sum _{i\in I}⟨u_i,v_i⟩$$

Linked PDF file:

http://images.planetmath.org/cache/objects/6363/pdf/DirectSumOfHilbertSpaces.pdf