weak topology

(redirected from Weak-* topology)

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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If X is a Banach space, the weak-* topology of the dual X' denoted by [sigma](X',X) is generated by the family {[p.sub.x]:x [member of] X} of seminorms, where [p.sub.x] (x') = |<x',x>|.
Remark 3.1.5 For a Banach space X, we consider in X' the weak-* topology [sigma](X',X).
If f: [OMEGA] [right arrow] R is a convex and continuous function, then x [right arrow] [partial derivative] f(x) of X in X' is a usco function, considering in [OMEGA] the norm topology and in X' the weak-* topology [sigma] (X',X).
The class [??] of Stegall spaces consists of all of the Banach spaces whose dual with the weak-* topology are topological [??]-spaces.
Since [V.sub.e] is isomorphic to [[??]'.sub.e] (when [[??].sub.e] is equipped with its formal topology previously introduced), it is the so-called weak-* topology. This topology turns [[??].sub.e] into a Hausdorff topological space (with K discrete).