Editing Direct sum of modules (section)

===<span id="Hilbertspaces"></span> Direct sum of Hilbert spaces===
{{further|Positive-definite kernel#Connection with reproducing kernel Hilbert spaces and feature maps}}

If finitely many [[Hilbert space]]s <math>H_1, \ldots, H_n</math> are given, one can construct their orthogonal direct sum as above (since they are vector spaces), defining the inner product as:
<math display=block>\left\langle \left(x_1, \ldots, x_n\right), \left(y_1, \ldots, y_n\right) \right\rangle = \langle x_1, y_1 \rangle + \cdots + \langle x_n, y_n \rangle.</math>

The resulting direct sum is a Hilbert space which contains the given Hilbert spaces as mutually [[orthogonal]] subspaces.

If infinitely many Hilbert spaces <math>H_i</math> for <math>i \in I</math> are given, we can carry out the same construction; notice that when defining the inner product, only finitely many summands will be non-zero. However, the result will only be an [[inner product space]] and it will not necessarily be [[Banach space|complete]]. We then define the direct sum of the Hilbert spaces <math>H_i</math> to be the completion of this inner product space. 

Alternatively and equivalently, one can define the direct sum of the Hilbert spaces <math>H_i</math> as the space of all functions α with domain <math>I,</math> such that <math>\alpha(i)</math> is an element of <math>H_i</math> for every <math>i \in I</math> and:
<math display=block>\sum_i \left\|\alpha_{(i)}\right\|^2 < \infty.</math>

The inner product of two such function α and β is then defined as:
<math display=block>\langle\alpha,\beta\rangle=\sum_i \langle \alpha_i,\beta_i \rangle.</math>

This space is complete and we get a Hilbert space.

For example, if we take the index set <math>I = \N</math> and <math>X_i = \R,</math> then the direct sum <math>\oplus_{i \in \N} X_i</math> is the space <math>\ell_2,</math> which consists of all the sequences <math>\left(a_i\right)</math> of reals with finite norm <math display="inline">\|a\| = \sqrt{\sum_i \left\|a_i\right\|^2}.</math> Comparing this with the example for [[Banach space]]s, we see that the Banach space direct sum and the Hilbert space direct sum are not necessarily the same. But if there are only finitely many summands, then the Banach space direct sum is isomorphic to the Hilbert space direct sum, although the norm will be different.

Every Hilbert space is isomorphic to a direct sum of sufficiently many copies of the base field, which is either <math>\R \text{ or } \Complex.</math>  This is equivalent to the assertion that every Hilbert space has an orthonormal basis.  More generally, every closed subspace of a Hilbert space is [[Complemented subspace|complemented]] because it admits an [[orthogonal complement]].  Conversely, the [[Lindenstrauss–Tzafriri theorem]] asserts that if every closed subspace of a Banach space is complemented, then the Banach space is isomorphic (topologically) to a Hilbert space.