Editing Direct sum of modules (section)

== Direct sum of modules with additional structure ==

If the modules we are considering carry some additional structure (for example, a [[Norm (mathematics)|norm]] or an [[inner product]]), then the direct sum of the modules can often be made to carry this additional structure, as well. In this case, we obtain the [[coproduct]] in the appropriate [[Category (category theory)|category]] of all objects carrying the additional structure. Two prominent examples occur for [[Banach space]]s and [[Hilbert space]]s.

In some classical texts, the phrase "direct sum of [[Algebra over a field|algebras over a field]]" is also introduced for denoting the [[algebraic structure]] that is presently more commonly called a [[direct product]] of algebras; that is, the [[Cartesian product]] of the [[underlying set]]s with the [[componentwise operation]]s. This construction, however, does not provide a coproduct in the category of algebras, but a direct product (''see note below'' and the remark on [[Direct sum#Direct sum of rings|direct sums of rings]]).

===Direct sum of algebras===

A direct sum of [[Algebra over a field|algebras]] <math>X</math> and <math>Y</math> is the direct sum as vector spaces, with product 
:<math>(x_1 + y_1) (x_2 + y_2) = (x_1 x_2 + y_1 y_2).</math>
Consider these classical examples:
:<math>\mathbf{R} \oplus \mathbf{R}</math> is [[Ring isomorphism|ring isomorphic]] to [[split-complex number]]s, also used in [[interval analysis]].
:<math>\mathbf{C} \oplus \mathbf{C}</math> is the algebra of [[tessarine]]s introduced by [[James Cockle (lawyer)|James Cockle]] in 1848.
:<math>\mathbf{H} \oplus \mathbf{H},</math> called the [[split-biquaternion]]s, was introduced by [[William Kingdon Clifford]] in 1873.
[[Joseph Wedderburn]] exploited the concept of a direct sum of algebras in his classification of [[hypercomplex number]]s. See his ''Lectures on Matrices'' (1934), page 151.
Wedderburn makes clear the distinction between a direct sum and a direct product of algebras: For the direct sum the field of scalars acts jointly on both parts: <math>\lambda (x \oplus y) = \lambda x \oplus \lambda y</math> while for the direct product a scalar factor may be collected alternately with the parts, but not both:
<math>\lambda (x,y) = (\lambda x, y) = (x, \lambda y).</math>
[[Ian R. Porteous]] uses the three direct sums above, denoting them <math>^2 R,\ ^2 C,\ ^2 H,</math> as rings of scalars in his analysis of ''Clifford Algebras and the Classical Groups'' (1995).

The construction described above, as well as Wedderburn's use of the terms {{em|direct sum}} and {{em|direct product}} follow a different convention than the one in [[category theory]]. In categorical terms, Wedderburn's {{em|direct sum}} is a [[Product (category theory)|categorical product]], whilst Wedderburn's {{em|direct product}} is a [[Coproduct|coproduct (or categorical sum)]], which (for commutative algebras) actually corresponds to the [[tensor product of algebras]].

=== 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>

===Direct sum of modules with bilinear forms===

Let <math>\left\{ \left(M_i, b_i\right) : i \in I \right\}</math> be a [[Indexed family|family]] indexed by <math>I</math> of modules equipped with [[bilinear form]]s. The '''orthogonal direct sum''' is the module direct sum with bilinear form <math>B</math> defined by<ref>{{cite book|first1=J.|last1=Milnor|author1-link=John Milnor|first2=D.|last2=Husemoller|title=Symmetric Bilinear Forms|series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]]|volume=73|publisher=[[Springer-Verlag]]|year=1973|isbn=3-540-06009-X|zbl=0292.10016|pages=4–5}}</ref>
<math display=block>B\left({\left({x_i}\right),\left({y_i}\right)}\right) = \sum_{i\in I} b_i\left({x_i,y_i}\right)</math>
in which the summation makes sense even for infinite index sets <math>I</math> because only finitely many of the terms are non-zero.

===<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.