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Locally Convex Topological Vector Spaces
Published in Kenneth Kuttler, Modern Analysis, 2017
The topology for X′ just described is called the weak* topology. In terms of Theorem 6.5 the weak topology is obtained by letting Y = X′ in that theorem while the weak* topology is obtained by letting Y = X with the understanding that X is a vector space of linear functionals on X′ defined by
Compact operators on Hilbert space
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
A very close result was given as Exercise 7.5.14. We note that these results are actually a special case of Alaoglu’s Theorem, which states that for every Banach space X, the closed unit ball (X*)1 of the dual space is compact in the so called weak-* topology.
Banach Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
The notion of the topological dual, understood as the space of all linear and continuous functionals, can be generalized to any topological vector space. Keeping this in mind, we could speculate what the topological dual corresponding to the weak topology would look like. As the weak topology is weaker than the strong topology, functionals continuous in the weak topology are automatically continuous in the strong topology. Surprisingly enough, the converse is also true: any strongly continuous and linear functional is also weakly continuous. This follows from the definition of the weak topology. Assume that U is a normed space, and f∈U′ $ f \in U^\prime $ , i.e., f is linear and continuous (in norm topology). In order to demonstrate that f is also continuous in the (corresponding) weak topology, we need to show that A∀ϵ>0∃B(I0,δ):u∈B(I0,δ)⇒|f(u)|<ϵ $$ A\forall \epsilon> 0 \, \exists B(I_0,\delta ) \, : \, u \in B(I_0,\delta ) \, \Rightarrow \, \vert f(u) \vert < \epsilon $$
A new approach to strong duality for composite vector optimization problems
Published in Optimization, 2021
María J. Cánovas, Nguyen Dinh, Dang H. Long, Juan Parra
Let X be a lcHtvs, whose origin is denoted by , and with topological dual space represented by . The only topology we consider on dual spaces is the weak*-topology. For a set , we denote by , , , and the closure, the linear hull, the interior, and the relative interior of U, respectively.
Cyclically antimonotone vector equilibrium problems
Published in Optimization, 2018
In general, for a Banach space X, we know by the Banach-Alaoglu Theorem (see, for instance, [21]) that the closed unit ball in is compact in weak* topology. Furthermore, if X is a reflexive Banach space, then we get as a consequence of Banach-Alaoglu Theorem that the closed unit ball in X is weakly compact. In this section, we suppose that X is a reflexive Banach space (the metric is induced by the norm of X) and with this setting, in the sequel we will use the following definitions.
New representations of epigraphs of conjugate mappings and Lagrange, Fenchel–Lagrange duality for vector optimization problems
Published in Optimization, 2023
Let X, Y, Z be locally convex Hausdorff topological vector spaces with their topological dual spaces denoted by and , respectively. The only topology we consider on dual spaces is the weak*-topology. For a set , we denote by , and the interior, the closure, the boundary, the convex hull, the linear hull, the affine hull, and the cone hull of U, respectively. For each , denotes the collection of all neighbourhoods of x in X. Assume that W is a topological subspace of X. For , denote by the interior of A w.r.t. the topology induced in W. Let K be a proper, closed, and convex cone in Y with nonempty interior, i.e. . It is worth observing that Weak Ordering Generated by a Convex Cone: Weak Infima and Weak Suprema. We define a weak ordering in Y generated by K as follows: for all , In Y we sometimes also consider an usual ordering generated by the cone K, , which is defined by if and only if for .