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In this paper, we study weak topology on right quaternionic Hilbert spaces. We show that on right quaternionic Hilbert spaces weak compactness and weak sequential compactness coincide which could be viewed as a generalization of Eberlein–Smulian theorem. Also, we apply this observation on study compact operators on right quaternionic Hilbert spaces.

In this paper we prove a result of the Trotter–Kato type in the weak topology. Let { A e } e > 0 be a family of quasi m-accretive linear operators on a Hilbert space X and let us denote by J λ e the resolvent of A e . Under certain conditions, the result states that if for any x ∈ X and k = 1 , 2 , … , the sequence ( J λ e ) k x converges weakly to ( J λ ) k x as e → 0 , where J λ is the resolvent of a linear quasi m-accretive operator A on X, then the sequence of the semigroups generated by − A e tends weakly to the semigroup generated by −A, uniformly with respect to t on compact intervals. The result is different from other results of the same type (see e.g., Yosida (1980) [9, p. 269] ) and gives an answer to an open problem put in Eisner and Sereny (2010) [3] . It is finally applied to compare the asymptotic behavior of a singular perturbation problem associated to a first order hyperbolic problem with diffusion.

We show that the unit ball of l p (Γ) in its weak topology is a continuous image of σ 1 (Γ) N , and we deduce some combinatorial properties of its lattice of open sets which are not shared by the balls of other equivalent norms when r is uncountable.

(2006). Curiosities concerning Weak Topology in Hilbert Space. The American Mathematical Monthly: Vol. 113, No. 5, pp. 447-452.

In this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and an Rδ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.

In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems.

Let be a real Hilbert space, a closed convex subset of , and continuous convex and bounded from below on . We study the asymptotic behaviour of the non-autonomous differential inclusion with project operator as follows where is an absolutely continuous decreasing control function with strictly positive value and . On the one hand, when , each solution of PS with initial point (not necessarily in ) converges to an element of in the weak topology. On the other hand, when and , the weak and strong convergence of PS to the minimizer of over can be guaranteed under some conditions. Strong convergence can be obtained when or is strongly convex on . Another two sufficient conditions are given to ensure the weak convergence. Moreover, asymptotic analysis for the special case that is also considered. Finally, we present some discussion on the global existence of the solutions of PS.

A lower hemi-continuous correspondence with open and convex values in Rn must have open lower sections. This well- known fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Since there is an increasing number of economic models that use correspondences in an infinite-dimensional setting, it is important to know whether or not the above fact remains valid in such applications. The aim of this paper is to show that the above fact no longer holds when Rn is replaced with an infinite-dimensional space. This is accomplished by using the standard orthonormal base in a Hilbert space H to construct two correspondences with values in H equipped with the weak topology. The first correspondence is lower hemi-continuous with open and convex values but does not have open lower sections. The second is a lower hemi-continuous correspondence that fails to have an open graph despite having open and convex upper and lower sections. These counter-examples demonstrate that in an infinite-dimensional setting, it is no longer possible to rely on the geometric properties of a lower hemi-continuous map (the convexity of its sections) to establish the topological properties (open lower sections, open graph) needed in many economic applications.

We study the computational difficulty of the problem of finding fixed points of nonexpansive mappings in uniformly convex Banach spaces. We show that the fixed point sets of computable nonexpansive self-maps of a nonempty, computably weakly closed, convex and bounded subset of a computable real Hilbert space are precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allows us to determine the Weihrauch degree of the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is equivalent to a closed choice principle, which receives as input a closed, convex and bounded set via negative information in the weak topology and outputs a point in the set, represented in the strong topology. While in finite dimensional uniformly convex Banach spaces, computable nonexpansive mappings always have computable fixed points, on the unit ball in infinite-dimensional separable Hilbert space the Browder-Goehde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive mappings may not have any computable fixed points in infinite dimension. We also study the computational difficulty of the problem of finding rates of convergence for a large class of fixed point iterations, which generalise both Halpern- and Mann-iterations, and prove that the problem of finding rates of convergence already on the unit interval is equivalent to the limit operator., 44 pages

In topological dynamics a theory of recurrence properties via (Fursten- berg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family F (of subsets of Z+) and a MDS (X,B,µ,T), several notions of ergodicity re- lated to F are introduced, and characterized via the weak topology in the induced Hilbert space L 2 (µ).

This paper deals with optimal feedback control of measure-valued solutions of nonlinear diffusion governed by McKean-Vlasov equations. Questions of existence, uniqueness, and regularity properties of measure-valued solutions are addressed. A class of feedback controls furnished with a weak topology is introduced, and some important topological properties of the attainable set corresponding to these controls are presented. We consider several typical control problems with objective functionals which are functions of measures and prove the existence of optimal controls.

Functional and convex analysis are closely intertwined. In this chapter we recall the basic concepts and results from functional analysis and calculus that will be needed throughout this book. A first section is devoted to general normed spaces. We begin by establishing some of their main properties, with an emphasis on the linear functions between spaces. This leads us to bounded linear functionals and the topological dual. Second, we review the Hahn–Banach Separation Theorem, a very powerful tool with important consequences. It also illustrates the fact that the boundaries between functional and convex analysis can be rather fuzzy at times. Next, we discuss some relevant results concerning the weak topology, especially in terms of closedness and compactness. Finally, we include a subsection on differential calculus, which also provides an introduction to standard smooth optimization techniques. The second section deals with Hilbert spaces, and their very rich geometric structure, including the ideas of projection and orthogonality. We also revisit some of the general concepts from the first section (duality, reflexivity, weak convergence) in the light of this geometry.

We study extension operators between spaces $\sigma_n(2^X)$ of subsets of $X$ of cardinality at most $n$. As an application, we show that if $B_H$ is the unit ball of a nonseparable Hilbert space $H$, equipped with the weak topology, then, for any $0

A characterisation of a type of weak-operator continuous linear functional on certain linear subsets of B(H), where H is a Hilbert space, is derived within Bishop-style constructive mathematics.

In this chapter, we investigate the existence and the topological structure of solution sets for fractional stochastic evolution inclusions in Hilbert spaces.

This introduction explains very briefly how and why the theory of Hilbert spaces and their operators emerged. In particular we mention which type of mathematical problems motivated the emergence of these spaces at the beginning of the 20th century as a generalization of finite linear vector spaces that have an inner product and therefore also metric and geometric properties. Then we present a short overview of the contents of this second part on Hilbert spaces and their operators. The final part of this introduction contains remarks on the particular and fundamental role which quantum physics has played in the development of the theory of Hilbert spaces and of modern functional analysis.

(i) Definitions, continuous linear functionals. Recall that a Hilbert space \( \mathcal{H} \) is a complex vector space equipped with an “inner product” (x|y) satisfying the following conditions.

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kac's model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space.

Stochastic approximation in a Hilbert space is considered. The main objective is to develop asymptotic theory for a computable finite dimensional approximation scheme. Nonlinear operators are treated, and a sequence of monotone projection operators is employed in the recursive algorithm. Under suitable conditions, convergence with respect to the weak topology is obtained

In Chapter 1, we have analyzed the structure of pip-spaces from the algebraic point of view only, (i.e., the compatibility relation). Here we will discuss primarily the topological structure given by the partial inner product itself. The aim is to tighten the definitions so as to eliminate as many pathologies as possible. The picture that emerges is reassuringly simple: Only two types of pip-spaces seem sufficiently regular to have any practical use, namely lattices of Hilbert spaces (LHS) or Banach spaces (LBS), that we have introduced briefly in the Introduction. Our standard reference on locally convex topological vector spaces (LCS) will be the textbook of Kothe [Kot69]. In addition, for the convenience of the reader, we have collected in Appendix B most of the necessary, but not so familiar, notions needed in the text. Notice that we diverge from [Kot69] for the notation. Namely, for a given dual pair \( \), we denote the weak topology on E by σ(E,F), its Mackey topology by t(E,F), and its strong topology by β(E,F) (see Appendix B).