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Linear Topological Spaces
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
The notion of linear topological spaces or topological vector spaces is very general and of great importance in functional analysis. It combines the purely topological concepts of topological spaces with the purely algebraic concepts of linear or vector spaces in a natural way such that the vector space operations of addition and multiplication by scalars are continuous.
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 $$ A\forall \epsilon> 0 \, \exists B(I_0,\delta ) \, : \, u \in B(I_0,\delta ) \, \Rightarrow \, \vert f(u) \vert < \epsilon $$
Ekeland variational principles for set-valued functions with set perturbations
Published in Optimization, 2020
Let Y be a topological vector space and let be its topological dual. Even though Y is Hausdorff, it still may happen that , i.e. there is no nontrivial continuous linear functional. For example, , is a topological vector space. But, every continuous linear functional on , vanishes identically, i.e. (see [45, p. 157–158]). However, if Y is a locally convex Hausdorff topological vector space (briefly, denoted by a locally convex space), then is large enough so that it separates points in Y, i.e. for any two different points in Y, there exists such that (for details, see [45–47]). For any , we define a continuous semi-norm on Y as follows: The semi-norm family generates a locally convex Hausdorff topology on Y (see, e.g. [45–47]), which is called the weak topology on Y and denoted by . For any nonempty subset F of , the semi-norm family can also generate a locally convex topology (needn't be Hausdorff) on Y, which is denoted by . In [46], the topology is called the F-projective topology.
On totalisation of computable functions in a distributive environment
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
In recent times, totalisation had remained a guiding principle towards interesting results in mathematics and sciences. One prominent example is the Hahn–Banach theorem, which is a principal tool in functional analysis having various applications to differential equations, characterisation of Banach spaces, and topological vector spaces [3,4]. It states the existence of the totalisation of linear functionals in real linear spaces.