weak topology


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weak topology

[′wēk tə′päl·ə·jē]
(mathematics)
A topology on a topological vector space X whose open neighborhoods around a point x are obtained from those points y of X for which every ƒi (x) is close to ƒi (y), ƒi appearing in a finite list of linear functionals.
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A Banach space X is a Stegall space if for every Baire space Z, every usco minimal function f: Z [right arrow] X' (with the weak topology) is single-valued in a residual subset.
In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. As long as the Banach space is reflexive, the weak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [1, 2].
Then F has compact closure in the weak topology [[tau].sup.[omega]] if and only if F is equiintegrable; that is,
This paper is devoted to presenting general results and examples for the existence of the fractional integral (and corresponding fractional differential) operators in arbitrary Banach space where it is endowed with its weak topology. In our investigations, we show that the well-known properties of the fractional calculus for functions taking values in finite dimensional spaces also hold in infinite dimensional spaces.
The weak topology on the space of all Borel measures M(X) is the weakest topology for which [[mu].sub.[alpha]] [right arrow] [mu] if and only if [integral] fd[[mu].sub.[alpha]] [right arrow] [integral] fd[mu] for every [integral] [integral] [C.sup.0](X).
Suppose X is a Banach space equipped with the norm topology (denoted by ||*||) as well as the weak topology (denoted by [T.sub.w]).
Similarly, by considering density in the weak topology instead of the norm topology, we can define weak hypercyclicity and weak supercyclicity.
Then we can, respectively, construct a random attractor endowed with the weak topology for the continuous random dynamical system in [V.sub.1] and [V.sub.2].
Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory Under Weak Topology for Nonlinear Operators and Black Operator Matrices With Applications
If K is a weakly compact subset of E and K with the relative weak topology is metrizable (for example E could be a Banach space whose dual [E.sup.*] is separable) then (2.2) holds (recall compact metric spaces are separable).
Since [L.sup.2]([??], dx) is densely embedded in S'([??]) with respect to the weak topology, see [3], then there exists a sequence [([[phi].sub.n]).sub.n] [member of] [L.sup.2]([??]) such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] weakly.
(a) [mathematical expression not reproducible], as j [right arrow] [infinity], in the weak topology on [L.sup.2](R, [gamma]),