Orthogonal sum

The three-dimensional Euclidean space can be represented as the orthogonal sum of a plane of origin and an orthogonal straight line through the origin. Each vector of space is then the sum of a part in the plane and a part on the straight line. ${\displaystyle V=U\oplus U^{\perp }}$ ${\displaystyle v=u+u^{\perp }}$

In the mathematical branch of linear algebra , the orthogonal sum is a construction that forms a single scalar product space, the orthogonal sum of the family, from a family of scalar product spaces (or more general spaces), in which the scalar product spaces can be embedded as pairwise orthogonal subspaces. The orthogonal sum is, so to speak, the minimum possible of such constructions. The scalar product defined on this space is also called the orthogonal sum or direct sum of the individual scalar products on the individual spaces. In functional analysis this construction is transferred to Hilbert spaces and then also speaks of the ( direct ) Hilbert sum or Hilbert space sum .

Outer orthogonal sum

Finite sums

First one considers two scalar product spaces and over the same field with the scalar products or . The outer orthogonal sum ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle K}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{X}}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{Y}}$

${\displaystyle X\oplus Y}$

is then the outer direct sum of the two vector spaces and , that is, the Cartesian product of the sets and with component addition and scalar multiplication . This space is now with the scalar product ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle X\times Y}$ ${\displaystyle X}$ ${\displaystyle Y}$

${\displaystyle \langle \,(x_{1},x_{2}),(y_{1},y_{2})\,\rangle _{X\oplus Y}=\langle x_{1},y_{1}\rangle _{X}+\langle x_{2},y_{2}\rangle _{Y}}$

for provided, the orthogonal sum of and . By means of the embeddings ${\displaystyle (x_{1},x_{2}),(y_{1},y_{2})\in X\oplus Y}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{X}}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{Y}}$

${\displaystyle x\mapsto (x,0)}$   and   ${\displaystyle y\mapsto (0,y)}$

can be identified with the sub-vector space and with , where the respective zero vector space is. A vector is then simply written as a sum , insofar as one o. B. d. A. can assume that the spaces are disjoint . If the vector spaces and over or are defined and complete , i.e. Hilbert spaces , then the space is also complete with regard to the scalar product . Orthogonal sums for a finite number of summands can also be defined inductively. ${\displaystyle X}$ ${\displaystyle X\times \{0\}}$ ${\displaystyle Y}$ ${\displaystyle \{0\}\times Y}$ ${\displaystyle \{0\}}$ ${\displaystyle (x,y)\in X\oplus Y}$ ${\displaystyle x+y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle X\oplus Y}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{X\oplus Y}}$

The orthogonal sum is also written as , for example, if other sums also occur. ${\displaystyle \oplus _{2}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle \oplus _{p}}$

Any direct sums

Let be a family of inner product spaces over the same field with inner products for an arbitrary index set . The direct sum of the vector spaces is the vector space ${\displaystyle (X_{i})_{i\in I}}$ ${\displaystyle K}$ ${\displaystyle \langle \cdot ,\cdot \rangle _{X_{i}}}$ ${\displaystyle I}$ ${\displaystyle X_{i}}$

${\displaystyle \bigoplus _{i}X_{i}=\left\{S\colon I\to \bigcup _{i}X_{i}\mid \forall i:S_{i}\in X_{i},\ \left\{i\in I\mid S(i)\neq 0\right\}{\text{ endlich}}\right\}}$

provided with the component-wise addition and scalar multiplication. The scalar product is defined on this space of families of vectors

${\displaystyle \langle \,(x_{i}),(y_{i})\,\rangle _{\bigoplus _{i}X_{i}}=\sum _{i\in I}\langle x_{i},y_{i}\rangle _{X_{i}}}$

with , which is well-defined , since according to the construction of the direct sum only finitely many of the summands are not equal to zero. This gives the (algebraic) orthogonal direct sum. By means of the embeddings ${\displaystyle x_{i},y_{i}\in X_{i}}$

${\displaystyle \mu _{j}\colon X_{j}\to \bigoplus _{i}X_{i},x_{j}\mapsto (x_{i})_{i\in I}}$   with   , ${\displaystyle x_{i}={\begin{cases}x_{j}&{\text{für}}\,\,i=j\\0&{\text{sonst}}\end{cases}}}$

get the scalar products, the individual spaces can again be identified with sub-vector spaces and a vector of this space can be simply written as a sum , although only a finite number of summands may differ from zero. The orthogonal sum of an empty family is the zero vector space (provided with the trivial and only possible scalar product). This construction is completely analogous for any families of modules over the same not necessarily commutative ring provided with any sesquilinear forms as the sesquilinear form on the direct sum. The direct sum is also defined for sesquilinear forms whose first and second arguments come from different modules, but in this case one can no longer speak of orthogonality of the embedded sub-modules. ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle \textstyle \sum _{i\in I}x_{i}}$

Arbitrary sums of Hilbert spaces

For such an infinite direct sum it generally no longer holds that the sum of Hilbert spaces is in turn a Hilbert space, so completeness can be violated. Therefore, for a family of Hilbert spaces over the same field ( or ) with scalar products, the orthogonal sum (or, clearly speaking, the Hilbert space sum) is defined as the completion of the above orthogonal direct sum (also called algebraic for delimitation ). In a sense, this is the smallest Hilbert space that contains the algebraic orthogonal direct sum. This is also called . Specifically, this space can be constructed as follows: ${\displaystyle (H_{i})_{i\in I}}$ ${\displaystyle K}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \langle \cdot ,\cdot \rangle _{H_{i}}}$ ${\displaystyle \bigoplus _{i}H_{i}}$

${\displaystyle \bigoplus _{i}H_{i}=\left\{S\colon I\to \bigcup _{i}H_{i}\mid \forall i:S_{i}\in H_{i},\ \left(\|S_{i}\|_{H_{i}}\right)_{i}\in \ell ^{2}(I)\right\}=\left\{S\colon I\to \bigcup _{i}H_{i}\mid \forall i:S(i)\in H_{i},\ \sum _{i}\|S(i)\|_{H_{i}}^{2}<\infty \right\}}$,

where the finiteness of the sum is to be read in such a way that, in particular, only at most countably many summands are not equal to zero. Addition and scalar multiplication are again explained component by component. The scalar product is again defined as

${\displaystyle \langle \,(x_{i}),(y_{i})\,\rangle _{\bigoplus _{i}H_{i}}=\sum _{i\in I}\langle x_{i},y_{i}\rangle _{H_{i}}}$,

whereby the definition now only ensures that only countably many summands are not equal to zero. The sum is to be read as an absolutely convergent series . As before, the embeddings provide identifications with Unterhilbert spaces and one writes a vector in the orthogonal sum, if necessary, simply as a sum , whereby now it only has to apply that the norms of the square sums are countable, so there can also be an infinite number of non-zero summands. ${\displaystyle \mu _{i}}$ ${\displaystyle (x_{i})_{i\in I}}$ ${\displaystyle \textstyle \sum _{i\in I}x_{i}}$ ${\displaystyle x_{i}}$

The construction as a completion shows that the (algebraic) direct sum is a dense sub-vector space of the orthogonal (Hilbert space) sum, which in turn is a sub-vector space of the direct product . In the case of an infinite family of spaces with no null vector spaces, these inclusions are real.

If confusion with the (algebraic) direct sum of vector spaces is possible, the orthogonal sum is also written as . As a special case of a sum, it is written as . ${\displaystyle \textstyle {\overline {\bigoplus _{i}H_{i}}}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle \textstyle \left(\bigoplus _{i}H_{i}\right)_{2}}$

Inner orthogonal sum

Analogous to the inner direct sum of vector spaces, in the special case of the orthogonal sum of pairwise orthogonal sub-Hilbert spaces of a given Hilbert space, one speaks of an inner orthogonal sum . While the condition of pairwise orthogonality is set for the inner orthogonal sum, an outer orthogonal sum can also be formed from many equal spaces, which are then "copied". If one considers a scalar product space, then the inner orthogonal sum of subspaces is nothing other than their inner direct vector space sum, i. H. their linear envelope .

The inner orthogonal (Hilbert space) sum in a Hilbert space, on the other hand, is the end of the linear envelope of the summands. It can easily be characterized by orthogonal projections : Let pairwise orthogonal Unterhilbert spaces of a Hilbert space , i.e. H. for and is . Then the orthogonal projections exist on the Unterhilbert spaces and their sum ${\displaystyle (H_{i})_{i\in I}}$ ${\displaystyle H}$ ${\displaystyle i\neq j}$ ${\displaystyle x\in H_{i},y\in H_{j}}$ ${\displaystyle x\perp y}$ ${\displaystyle \pi _{i}\colon H\to H_{i}}$

${\displaystyle \pi \colon H\to H,x\mapsto \sum _{i}\pi _{i}(x)}$.

is again an orthogonal projection. The image of is just the (inner) orthogonal sum of the spaces . ${\displaystyle \pi }$ ${\displaystyle H_{i}}$

Examples

• Space is the special case of the orthonormal sum of the one-dimensional Hilbert space : ${\displaystyle \ell ^{2}(I)}$ ${\displaystyle K}$

${\displaystyle \ell ^{2}(I)=\bigoplus _{i\in I}K}$

• For a negative Hilbert space is just the sum of inner orthogonal and its orthogonal complement : ${\displaystyle U\subseteq H}$ ${\displaystyle H}$ ${\displaystyle U}$ ${\displaystyle U^{\perp }}$

${\displaystyle H=U\oplus U^{\perp }}$

• The antisymmetric Fock space , which is important in quantum field theory, results as the completion of the external algebra on a Hilbert space , or as the orthogonal sum of the external powers : ${\displaystyle H}$ ${\displaystyle \Lambda ^{i}(H)}$

${\displaystyle F_{-}(H)=\bigoplus _{i\in \mathbb {N} }\Lambda ^{i}(H)}$

• Correspondingly, the symmetrical Fock space results as the completion of the symmetrical algebra , another quotient of the tensor algebra , or as the orthogonal sum of the spaces of the symmetrical tensors of the level above : ${\displaystyle S^{i}(H)}$ ${\displaystyle i}$ ${\displaystyle H}$

${\displaystyle F_{+}(H)=\bigoplus _{i\in \mathbb {N} }S^{i}(H)}$

Bases and dimension

Let be orthonormal bases of . Then is an orthonormal basis of . This union is disjoint because the embedded sub-helper spaces are orthogonal in pairs and a base element is never zero. Thus the Hilbert space dimension of the orthogonal sum is equal to the sum of the dimensions of the individual Hilbert spaces: ${\displaystyle B_{i}\subset H_{i}}$ ${\displaystyle H_{i}}$ ${\displaystyle \textstyle \bigcup _{i}\mu _{i}(B_{i})}$ ${\displaystyle \textstyle \bigoplus _{i}H_{i}}$

${\displaystyle \dim \bigoplus _{i}H_{i}=\left|\bigcup _{i}\mu _{i}(B_{i})\right|=\sum _{i}\left|B_{i}\right|=\sum _{i}\dim H_{i}}$.

In particular, or more generally, applies to cardinal numbers . ${\displaystyle \mathbb {R} ^{n}\oplus \mathbb {R} ^{m}\cong \mathbb {R} ^{n+m}}$ ${\displaystyle \ell ^{2}(\alpha )\oplus \ell ^{2}(\beta )\cong \ell ^{2}(\alpha +\beta )}$ ${\displaystyle \alpha ,\beta }$

Categorical properties

In the algebraic case of the orthogonal sum of scalar product spaces or sesquilinear forms on modules, the orthogonal sum of the spaces is nothing else than the direct sum, the individual scalar products have no influence on the structure of this space. This is a coproduct in the corresponding category of modules with linear mapping. The finite direct sum is also a (direct) product , while in the infinite case it generally differs from this.

The orthogonal sum of finitely many Hilbert spaces is analogously a biproduct in the category of Hilbert spaces with continuous linear operators as morphisms , i.e. H. it is both a (direct) product and a co-product (direct sum). In addition, this biproduct is compatible with the structure that is given by the adjoint that ${\displaystyle \dagger }$

${\displaystyle \mu _{i}^{\dagger }=\pi _{i}}$ and

${\displaystyle \pi _{i}^{\dagger }=\mu _{i}}$.

Since the null morphisms (i.e., specifically the null functions ) apply to any Hilbert space , one speaks of a biproduct category . ${\displaystyle (0\colon X\to Y)^{\dagger }=(0\colon Y\to X)}$ ${\displaystyle X,Y}$ ${\displaystyle \dagger }$

In contrast, the orthogonal sum of an infinite family of non-zero spaces in this category is neither a product nor a coproduct: To see that it is not a product, consider the morphisms for unit vectors . If there were a product with the projections , there would have to be a continuous linear mapping with , i. H. would have the amount in each component with which there would no longer be any square summability. Dual to consider for the violation of the coproduct property consider o. B. d. A. (for uncountable choose surplus as zero) and the morphisms . Now a morphism should exist with . However , this could at best be unlimited and only densely defined , because for ${\displaystyle v_{i}\in H_{i}}$ ${\displaystyle f_{i}\colon K\to H_{i},\lambda \mapsto \lambda v_{i}}$ ${\displaystyle \textstyle \bigoplus _{i}H_{i}}$ ${\displaystyle \pi _{i}}$ ${\displaystyle \textstyle f\colon K\to \bigoplus _{i}H_{i}}$ ${\displaystyle f_{i}=\pi _{i}f}$ ${\displaystyle f(1)}$ ${\displaystyle 1}$ ${\displaystyle I=\mathbb {N} \setminus \{0\}}$ ${\displaystyle I}$ ${\displaystyle g_{i}}$ ${\displaystyle g_{i}\colon H_{i}\to K,u\mapsto i\cdot \langle u,v\rangle _{H_{i}}}$ ${\displaystyle \textstyle g\colon \bigoplus _{i}H_{i}\to K}$ ${\displaystyle g_{i}=g\mu _{i}}$ ${\displaystyle g}$

${\displaystyle u=\sum _{i}{\frac {1}{i}}v_{i}\in \bigoplus _{i}H_{i}}$

would have to

${\displaystyle g(u)=\sum _{i}i\cdot \langle {\frac {1}{i}}v_{i},v_{i}\rangle =\sum _{i}1}$

be what, however, diverges . In fact, there are no ( small ) products or coproducts in this category . The examples also show that many conceivable restrictions of the morphisms do not provide a remedy - the example morphisms are of rank one and therefore very benign. The choice of linear contractions (which in the case of the Banach spaces leads to the completeness of the category and in the case of the Hilbert spaces fixes unitary operators as the isomorphisms ) is also not possible; in this case the orthogonal sum would no longer even be a finite product or co-product. It is not possible to resort to tightly defined operators, as these are not concluded under composition and therefore do not form a category.

See also

• The orthogonal sum is the special case of the ℓ ^p -sum of Banach spaces for . ${\displaystyle p=2}$
• The orthogonal sum is also a special case of the direct integral of Hilbert spaces with the counting measure .
• The Hilbert space tensor product is another important construction on Hilbert spaces.

Individual evidence

1. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-35338-0 , chap. 9 , p. 13 .
2. ^ Nicolas Bourbaki : V. Topological Vector Spaces (=  Elements of Mathematics ). Springer , Berlin 2003, ISBN 3-540-42338-9 , V, p. 17 (Original title: Éspaces vectoriels topologiques . Paris 1981. Translated by HG Eggleston and S. Madan).
3. ↑ John Harding, Orthomodularity in dagger biproduct categories (PDF; 301 kB) , 2008, p. 5
4. ↑ Chris Heunen, An Embedding Theorem for Hilbert Categories (PDF; 275 kB) , in Theory and Applications of Categories , 2009, p. 339