Shudder base

In functional analysis , a sequence of a Banach space is called the shudder basis if every vector has a unique representation as an (infinite) linear combination with respect to it. It is to be differentiated from the Hamel basis , from which it is required that every vector can be represented as a finite linear combination of the basis elements. ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$

The Schauderbasen are named after the Polish mathematician Juliusz Schauder (1899–1943), who described them in 1927.

definition

Be a Banach space above the basic body or . A sequence in is called a shudder basis if each can be clearly represented as a convergent series . ${\displaystyle (X,\left\|\cdot \right\|)}$ ${\displaystyle \mathbb {K} =\mathbb {R} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle x\in X}$ ${\displaystyle \textstyle x=\sum _{n=1}^{\infty }\xi _{n}\cdot b_{n},\;\xi _{n}\in \mathbb {K} }$

Examples

• In the sequence space with the ℓ ^p -norm form a shudder basis for the unit vectors . ${\displaystyle \textstyle \ell ^{p}:=\left\{(x_{j})_{j=1}^{\infty },x_{j}\in \mathbb {R} \,:\,\sum _{j=1}^{\infty }|x_{j}|^{p}<\infty \right\}}$ ${\displaystyle \textstyle \left\|x\right\|_{\ell ^{p}}={\sqrt[{p}]{\sum _{j=1}^{\infty }|x_{j}|^{p}}}}$ ${\displaystyle 1\leq p<\infty }$ ${\displaystyle (1,0,0,\dotsc ),(0,1,0,0,\dotsc ),\dotsc }$

• Set for all , and for , defined by ${\displaystyle h_{1}(x)=1}$ ${\displaystyle x\in [0,1]}$ ${\displaystyle 1\leq i\leq 2^{n}}$ ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle h_{2^{n}+i}\colon [0,1]\rightarrow \mathbb {R} }$

${\displaystyle h_{2^{n}+i}(x)={\begin{cases}1,&(2i-2)/2^{n+1}\leq x<(2i-1)/2^{n+1},\\-1,&(2i-1)/2^{n+1}\leq x<2i/2^{n+1},\\0&{\mbox{sonst}}.\end{cases}}}$

With the exception of a constant factor, each is a Haar wavelet function that is restricted to one another. The sequence , which is also called the Haar system after Alfréd Haar , is a shudder basis for the space L ^p ([0,1]) for . ${\displaystyle h_{k}}$ ${\displaystyle [0,1)}$ ${\displaystyle (h_{k})_{k\in \mathbb {N} }}$ ${\displaystyle 1\leq p<\infty }$

• For the construction of a shudder basis of the room let a dense sequence in without repetitions and let it be . Take, for example, a bijective counting of the rational points of the unit interval or a sequence of the type and so on by continuously halving the gaps left by the sequence formed so far. ${\displaystyle C([0,1])}$ ${\displaystyle (q_{n})_{n\in \mathbb {N} }}$ ${\displaystyle [0,1]}$ ${\displaystyle q_{1}=0,q_{2}=1}$ ${\displaystyle 0,1,{\tfrac {1}{2}},{\tfrac {1}{4}},{\tfrac {3}{4}},{\tfrac {1}{8}},{\tfrac {3}{8}},{\tfrac {5}{8}},{\tfrac {7}{8}},\ldots ,{\tfrac {2k+1}{2^{n}}},\ldots }$

For each one is defined by = constant 1 and for all others let , for all and be affine-linear on . Then the consequence is a shudder basis of C ([0,1]). The idea for the construction of this shudder base goes back to Juliusz Schauder and one calls such a base the shudder base. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle e_{n}\in C([0,1])}$ ${\displaystyle e_{1}}$ ${\displaystyle n>1}$ ${\displaystyle e_{n}(q_{n})=1}$ ${\displaystyle e_{n}(q_{k})=0}$ ${\displaystyle k=1,\ldots ,n-1}$ ${\displaystyle e_{n}}$ ${\displaystyle [0,1]\setminus \{q_{1},\ldots ,q_{n}\}}$ ${\displaystyle (e_{n})_{n\in \mathbb {N} }}$

properties

General properties

• A Banach space with a shudder basis is separable , because the set of finite linear combinations with coefficients of or is a dense, countable set. ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {Q} +i\mathbb {Q} }$
• Conversely, not every separable Banach space has a shudder base.
• Banach spaces with a shudder basis have the approximation property .
• In infinite-dimensional Banach spaces, a horror basis is never the Hamel basis of the vector space, since such a basis must always be uncountable in infinite-dimensional Banach spaces (see Baire's theorem ).

Coefficient functionals

The representation of an element in relation to a shudder base is clear by definition. The assignments are called coefficient functionals; they are linear and continuous and therefore elements of the dual space of . ${\displaystyle x\in X}$ ${\displaystyle b_{n}^{\ast }\colon x\mapsto \xi _{n}}$ ${\displaystyle X}$

Properties of the base

Shudder bases can have further properties. The existence of shudder bases with such properties then has further consequences for the Banach space.

If the Banach space is a shudder basis , there is a constant such that the inequality holds for and for every choice of scalars . The infimum over those that satisfy this inequality on a given basis is called the basis constant . One speaks of a monotonic basis if the basis constant is equal to 1. ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle K>0}$ ${\displaystyle p<q}$ ${\displaystyle \xi _{n}\in {\mathbb {K} }}$ ${\displaystyle \textstyle \left\|\sum _{n=1}^{p}\xi _{n}b_{n}\right\|\,\leq \,K\cdot \left\|\sum _{n=1}^{q}\xi _{n}b_{n}\right\|}$ ${\displaystyle K>0}$

This is called a base limited entirely (English: boundedly complete), if for every sequence of scalars with a are with . ${\displaystyle (b_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (\xi _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \textstyle \sup _{m\in \mathbb {N} }\left\|\sum _{n=1}^{m}\xi _{n}b_{n}\right\|<\infty }$ ${\displaystyle x\in X}$ ${\displaystyle \textstyle x=\sum _{n=1}^{\infty }\xi _{n}b_{n}}$

Furthermore, let be the closed subspace generated by , and let be the norm of the restricted functional . The base is called shrinking , if for everyone . ${\displaystyle X_{n}\subset X}$ ${\displaystyle (b_{j})_{j\geq n}}$ ${\displaystyle f\in X\,'}$ ${\displaystyle \|f|_{X_{n}}\|}$ ${\displaystyle f|_{X_{n}}\in X_{n}'}$ ${\displaystyle \lim _{n\to \infty }\|f|_{X_{n}}\|=0}$ ${\displaystyle f\in X\,'}$

Finally, one speaks of an unconditional basis if all series in the developments with regard to the basis necessarily converge . The standard bases of the spaces are obviously unconditional. The space does not have an unconditional basis. Using the property (u) of Pelczynski one can even show that he is not even a subspace of a Banach space with an unconditional basis. It can also be shown that the hair system is in for an unconditional basis, but not for . The room has no unconditional basis. ${\displaystyle \textstyle x=\sum _{n=1}^{\infty }\xi _{n}b_{n}}$ ${\displaystyle \ell ^{p}}$ ${\displaystyle C([0,1])}$ ${\displaystyle L^{p}([0,1])}$ ${\displaystyle 1<p<\infty }$ ${\displaystyle p=1}$ ${\displaystyle L^{1}([0,1])}$

Two sentences from RC James

The following two sentences by Robert C. James show the meaning of the basic terms.

• RC James: Be a Banach room with a shudder base. is reflexive if and only if the basis is restrictedly complete and shrinking. ${\displaystyle X}$ ${\displaystyle X}$

The presence of certain subspaces can be characterized for unconditional shudder bases. Be a Banach space with an unconditional shudder base. Then: ${\displaystyle X}$

• ${\displaystyle X}$contains no subspace isomorphic to c ₀ . The basis is limited and complete. ${\displaystyle \Leftrightarrow }$
• ${\displaystyle X}$does not contain a subspace that is too isomorphic. The base is shrinking. ${\displaystyle \ell ^{1}}$ ${\displaystyle \Leftrightarrow }$

The consequence of this is:

• RC James: Be a Banach room with an unconditional shudder base. is reflexive if and only if it contains no subspace that is too or isomorphic. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle c_{0}}$ ${\displaystyle \ell ^{1}}$

See also

• Base sequence

literature

• Bernard Beauzamy: Introduction to Banach Spaces and their Geometry , Elesevier Science Publishers (1985) ISBN 0-444-87878-5
• Zdzisław Denkowski, Stanisław Migórski, Nikolas S. Papageorgiou: An introduction to nonlinear analysis. Kluwer, Boston 2003, ISBN 0-306-47392-5
• Joseph Diestel: Sequences and Series in Banach Spaces. 1984, ISBN 0-387-90859-5 .
• Yuli Eidelman, Vitali Milman, Antonis Tsolomitis: Functional analysis. An introduction. American Mathematical Society, Providence 2004, ISBN 0-8218-3646-3
• Ivan Singer: Bases in Banach spaces I (1970) and Bases in Banach spaces II (1981), Springer Verlag

Individual evidence

1. ↑ F. Albiac, NJ Kalton: Topics in Banach Space Theory : Springer-Verlag (2006), ISBN 978-0-387-28142-1 , page 9f
2. ↑ Per Enflo : A counterexample to the approximation problem in Banach spaces. Acta Mathematica, Vol. 130, No. 1, July 1973, pp. 309-317