Direct sum (Banach space)

The direct sum of Banach spaces is a method used in the mathematical branch of functional analysis to construct new ones from given Banach spaces . The algebraic direct sums are given a suitable norm. This already turns finite direct sums into Banach spaces, with infinitely many summands one still has to complete .

Finite direct sums

Let them be standardized spaces whose norms are marked with . Then the direct sum is again a standardized space, if you refer to the norms ${\displaystyle X_{1},\ldots ,X_{n}}$ ${\displaystyle \|\cdot \|}$ ${\displaystyle X_{1}\oplus \ldots \oplus X_{n}}$

${\displaystyle \|(x_{1},\ldots ,x_{n})\|_{p}:=\left(\sum _{j=1}^{n}\|x_{j}\|^{p}\right)^{\frac {1}{p}},\quad \quad 1\leq p<\infty }$

or

${\displaystyle \|(x_{1},\ldots ,x_{n})\|_{\infty }:=\max _{j=1,\ldots ,n}\|x_{j}\|}$

explained. These norms are mutually equivalent and induce the product topology of the direct sum . With each of these norms, the direct sum is a Banach space if and only if each of them is a Banach space. In the case of Hilbert spaces one uses the above definition for , because only with this norm is the direct sum a Hilbert space again. ${\displaystyle X_{j}}$ ${\displaystyle X_{1}\oplus \ldots \oplus X_{n}}$ ${\displaystyle X_{j}}$ ${\displaystyle p=2}$

Infinite sums

Let it be a sequence of Banach spaces, where the norm is denoted on each with . Then you define ${\displaystyle (X_{j})_{j\in \mathbb {N} }}$ ${\displaystyle X_{j}}$ ${\displaystyle \|\cdot \|}$

${\displaystyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{p}=\left\{(x_{j})_{j}\in \prod _{j\in \mathbb {N} }X_{j};\,\sum _{j\in \mathbb {N} }\|x_{j}\|^{p}<\infty \right\}}$

and

${\displaystyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{\infty }=\left\{(x_{j})_{j}\in \prod _{j\in \mathbb {N} }X_{j};\,\sup _{j\in \mathbb {N} }\|x_{j}\|<\infty \right\}}$.

They are subspaces of the Cartesian product and the norms ${\displaystyle \prod _{j\in \mathbb {N} }X_{j}}$

${\displaystyle \|(x_{j})_{j}\|_{p}:=\left(\sum _{j\in \mathbb {N} }\|x_{j}\|^{p}\right)^{\frac {1}{p}}}$

or.

${\displaystyle \|(x_{j})_{j}\|_{\infty }:=\sup _{j\in \mathbb {N} }\|x_{j}\|}$

make them Banach spaces. Finally is

${\displaystyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{0}=\left\{(x_{j})_{j}\in \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{\infty };\,\|x_{j}\|\rightarrow 0\right\}}$

an underbench room.

If one chooses specifically or for all j , one obtains the known sequence spaces or . One calls therefore the sum of the Banach spaces, for or one speaks of the sum or sum. If the Banach spaces are all the same, roughly the same , the above spaces are written shorter than or . ${\displaystyle X_{j}=\mathbb {R} }$ ${\displaystyle X_{j}=\mathbb {C} }$ ${\displaystyle \ell ^{p},\ell ^{\infty }}$ ${\displaystyle c_{0}}$ ${\displaystyle \textstyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{p}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle p=\infty }$ ${\displaystyle p=0}$ ${\displaystyle \ell ^{\infty }}$ ${\displaystyle c_{0}}$ ${\displaystyle X}$ ${\displaystyle \ell ^{p}(X)}$ ${\displaystyle c_{0}(X)}$

If one denotes the dual space of a Banach space with , one has in analogy to the sequence spaces: ${\displaystyle X}$ ${\displaystyle X^{*}}$

${\displaystyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{0}^{*}=\left(\sum _{j\in \mathbb {N} }X_{j}^{*}\right)_{1}}$

and

${\displaystyle \left(\sum _{j\in \mathbb {N} }X_{j}\right)_{p}^{*}=\left(\sum _{j\in \mathbb {N} }X_{j}^{*}\right)_{q}}$   for and . ${\displaystyle 1\leq p<\infty }$ ${\displaystyle {\frac {1}{p}}+{\frac {1}{q}}=1}$

This dual space relationship is to be understood in such a way that a sequence is to be applied as a functional to a sequence using the formula and that the norm of the functional is exactly the norm in the corresponding sum of the dual spaces. ${\displaystyle \textstyle f=(f_{j})_{j}\in \prod X_{j}^{*}}$ ${\displaystyle x=(x_{j})_{j}}$ ${\displaystyle \textstyle f(x)=\sum _{j\in \mathbb {N} }f_{j}(x_{j})}$

The algebraic direct sum is usually not itself a Banach space, but it lies close to the -sum or in the -sums for , the latter are thus completions of the direct sum. In contrast to the situation of finite direct sums, the completions are not isomorphic with respect to the various norms. This is already shown by the example for all j , because then the completions are the -spaces which, according to Pitt's theorem, are not isomorphic with one another. ${\displaystyle \oplus _{j}X_{j}}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle X_{j}=\mathbb {R} }$ ${\displaystyle \ell ^{p}}$

In some cases, constructions of this type do not create new spaces, which is a useful property of these spaces. For example ${\displaystyle \ell ^{p}(X)}$

${\displaystyle \ell ^{p}(L^{p}[0,1])\cong L^{p}[0,1]}$ For ${\displaystyle 1\leq p\leq \infty }$

${\displaystyle \ell ^{p}(\ell ^{p})\cong \ell ^{p}}$ For ${\displaystyle 1\leq p\leq \infty }$

${\displaystyle c_{0}(c_{0})\cong c_{0}}$

${\displaystyle c_{0}(C(\Delta ))\cong C(\Delta )}$.

Here stands for isometric isomorphism, is the L ^p -space above the unit interval and is the space of the continuous functions on the Cantor space . ${\displaystyle \cong }$ ${\displaystyle L^{p}[0,1]}$ ${\displaystyle C(\Delta )}$

Individual evidence

1. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 1.8.1, there only for p = 2, the general case can be found in the exercises.
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 1.8.2
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , theorem 1.8.6
4. ↑ J. Lindenstrauss, L. Tzafriri: Classical Banach spaces I Springer, Berlin et al. 1977, ISBN 978-3-642-66559-2 , page xii: Standard Definitions, Notations and Conventions
5. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , Chapter 2.2
6. ↑ P. Wojtaszczyk: Banach Spaces for Analysts , Cambridge studies in advanced mathematics 25, ISBN 978-0-521-56675-9 II.B. 21st
7. ↑ P. Wojtaszczyk: Banach Spaces for Analysts , Cambridge studies in advanced mathematics 25, ISBN 978-0-521-56675-9 II.B. 21st