Editing Direct sum of modules (section)

=== Direct sum of Banach spaces ===
{{anchor|Banachspaces}}

The direct sum of two [[Banach space]]s <math>X</math> and <math>Y</math> is the direct sum of <math>X</math> and <math>Y</math> considered as vector spaces, with the norm <math>\|(x, y)\| = \|x\|_X + \|y\|_Y</math> for all <math>x \in X</math> and <math>y \in Y.</math>

Generally, if <math>X_i</math> is a collection of Banach spaces, where <math>i</math> traverses the [[index set]] <math>I,</math> then the direct sum <math>\bigoplus_{i \in I} X_i</math> is a module consisting of all functions <math>x</math> [[domain of a function|defined over]] <math>I</math> such that <math>x(i) \in X_i</math> for all <math>i \in I</math> and
<math display=block>\sum_{i \in I} \|x(i)\|_{X_i} < \infty.</math>

The norm is given by the sum above. The direct sum with this norm is again a Banach space.

For example, if we take the index set <math>I = \N</math> and <math>X_i = \R,</math> then the direct sum <math>\bigoplus_{i \in \N} X_i</math> is the space <math>\ell_1,</math> which consists of all the sequences <math>\left(a_i\right)</math> of reals with finite norm <math display="inline">\|a\| = \sum_i \left|a_i\right|.</math>

A closed subspace <math>A</math> of a Banach space <math>X</math> is '''[[complemented subspace|complemented]]''' if there is another closed subspace <math>B</math> of <math>X</math> such that <math>X</math> is equal to the internal direct sum <math>A \oplus B.</math> Note that not every closed subspace is complemented; e.g. [[c0 space|<math>c_0</math>]] is not complemented in <math>\ell^\infty.</math>