Limited weak - * topology

The restricted weak - * - topology , abbreviated bw * - topology (after the English term "bounded weak * topology"), is a topology examined in the mathematical subfield of functional analysis on the dual space of a standardized space . It is closely related to the weak - * topology .

definition

Be a normalized space and its dual space. The bw * topology is the finest topology on whose relative topology at all limited quantities with the weak - * - matches topology. ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X'}$

If the inclusion is defined for each restricted set , the bw * topology is the final topology of the mappings . A set is bw * -open if and only if the average for all bounded sets is relatively weak - * - open. ${\displaystyle B\subset X'}$ ${\displaystyle \iota _{B}:B\rightarrow X'}$ ${\displaystyle \iota _{B}}$ ${\displaystyle U\subset X'}$ ${\displaystyle U\cap B}$ ${\displaystyle B\subset X'}$

Basis of the bw * topology

The basis of the bw * topology described here goes back to Jean Dieudonné . If a normalized space, an element of the dual space and a null sequence in , then let ${\displaystyle X}$ ${\displaystyle f\in X'}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$

${\displaystyle B(f,(x_{n})_{n}):=\{g\in X';\,|(f-g)x_{n}|<1{\text{ für alle }}n\in \mathbb {N} \}}$.

These sets form an environment base of open sets of . Since these sets are obviously convex , the bw * topology is a locally convex Hausdorff topology. If there is a null sequence, then it is through ${\displaystyle f}$ ${\displaystyle (x_{n})_{n}}$

${\displaystyle p_{(x_{n})_{n}}(f):=\sup _{n\in \mathbb {N} }|f(x_{n})|}$

a semi-norm is defined and the bw * topology is exactly the locally convex topology generated by these semi-norms. ${\displaystyle X'}$

completeness

If a normalized space is, then the dual space with the bw * topology is complete , i.e. every bw * - Cauchy network converges. More precisely, this means: If a network is in , so that there is an index for every zero sequence from , so that for all , there is a with with respect to the bw * topology. ${\displaystyle X}$ ${\displaystyle (f_{\alpha })_{\alpha }}$ ${\displaystyle X'}$ ${\displaystyle (x_{n})_{n}}$ ${\displaystyle X}$ ${\displaystyle \gamma }$ ${\displaystyle f_{\alpha }-f_{\beta }\in B(0,(x_{n})_{n})}$ ${\displaystyle \alpha ,\beta >\gamma }$ ${\displaystyle f\in X'}$ ${\displaystyle f_{\alpha }\rightarrow f}$

In particular, the result is that the bw * topology for infinite-dimensional spaces is really finer than the weak - * topology, because the latter is known to be incomplete.

bw * -continuous linear functionals

Is a Banach * - - steady and bw * -stetigen, the weak fall linear functionals on together. This results in ${\displaystyle X}$ ${\displaystyle X'}$

• A linear functional on is weakly - * - continuous if and only if the restriction to the unit sphere is weak - * - continuous. ${\displaystyle X'}$

In addition, the Kerin-Šmulian theorem about weak - * - closed, convex sets can very easily be derived from this. This is detailed in the textbook given below.

Compact operators

Compact operators can be characterized using the bw * topology . If there is a continuous, linear operator between Banach spaces, the adjoint operator is known to be continuous if the norm topology, the weak - * - topology or the bw * - topology is considered on both spaces. Interesting statements can therefore only be expected if one considers different topologies in the rooms. The following sentence applies: ${\displaystyle T:X\rightarrow Y}$ ${\displaystyle T':Y'\rightarrow X'}$

• A continuous linear operator between Banach spaces is compact if and only if the adjoint operator is continuous with respect to the bw * topology on and the norm topology on . ${\displaystyle T:X\rightarrow Y}$ ${\displaystyle T':Y'\rightarrow X'}$ ${\displaystyle Y'}$ ${\displaystyle X'}$

bw topology and cbw topology

In analogy to the bw * topology on a dual space, the bw topology on the output space can be defined as the finest topology that corresponds to the relatively weak topology on all limited sets. This topology is far from having the same meaning as the bw * topology, because it is generally not locally convex. In 1974 RF Wheeler showed that the bw topology on the sequence space is not locally convex, and in 1984 J. Gómez Gil was even able to show that the bw topology is locally convex if and only if the space is reflexive . For reflexive spaces, however , the bw topology does not bring anything new, because then there is itself a dual space, and the bw topology agrees with the bw * topology if one identifies with . ${\displaystyle c_{0}}$ ${\displaystyle X}$ ${\displaystyle X\cong X''}$ ${\displaystyle X}$ ${\displaystyle X''}$

In order to obtain a locally convex topology, one defines the cbw topology, which is generated from all convex, open sets of the bw topology. This is locally convex and agrees with the relative bw * topology of , if one understands the canonical embedding as a subspace of . ${\displaystyle X}$ ${\displaystyle X''}$ ${\displaystyle X}$ ${\displaystyle X''}$

Individual evidence

1. ^ J. Dieudonné: Natural homomorphisms in Banach spaces , Proceedings American Mathematical Society (1950), Volume 1, pages 54-59
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 2.7.2
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 2.7.6 with corollary 2.7.7
4. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 2.7.8 - 2.7.11
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Theorem 3.4.16
6. ↑ RF Wheeler: The equicontinuous weak * topology and semi-reflexivity , Studia Mathematica (1972), Volume 41, pages 243-256
7. ^ J. Gómez Gil: On local convexity of bounded weak topologies on Banach spaces , Pacific J. Math. (1984), Volume 110, Number 1, Pages 71-76
8. ↑ JG Llavona: Approximation of Continuously Differentiable Functions , Elsevier Science Publishers (1986), ISBN 0-444-70128-1 , Definition 4.2.2, Theorem 4.2.3