On Almost β-Topological Vector Spaces

Let (X, =) be a topological space. A subset A of X is called pre-open if A ⊆ Int(Cl(A)). Let P O(X) denote the family of all pre-open sets in X. In general, P O(X) does not form a topology on X. Furthermore, in topological vector spaces, it is not always true that P O(L) forms a topology on L where L is a topological vector space. In this note, we prove that the class of strongly preirresolute topological vector spaces is that subclass of topological vector spaces in which P O(L) forms a topology and thereby we see that all proved results in [5] concerning strongly preirresolute topological vector spaces are obvious.

The spaces of test functions we are going to use are vector spaces of smooth (i.e., sufficiently often continuously differentiable) functions on open nonempty subsets Ω ⊆ ℝ n equipped with a ‘natural’ topology. Accordingly we start with a general method to equip a vector space V with a topology such that the vector space operations of addition and scalar multiplication become continuous, i.e., such that $$ \begin{array}{*{20}c} {_{M\,:\,\,\mathbb{K}\, \times \,V\, \to \,V,}^{A\,:\,\,V\, \times \,V\, \to \,V,} } & {_{M(\lambda ,\,x)\, = \,\lambda x,\,}^{A(x,\,y)\, = \,x\, + \,y,} } & {_{\lambda \, \in \,\mathbb{K},\,\,\,x\, \in \,V}^{x,\,y\, \in \,V} } \end{array} $$ become continuous functions for this topology. This can be done in several different but equivalent ways. The way we describe has the advantage of being the most natural one for the spaces of test functions we want to construct. A vector space V which is equipped with a topology T such that the functions A} and M are continuous is called a topological vector space, usually abbreviated as TVS. The test function spaces used in distribution theory are concrete examples of topological vector spaces where, however, the topology has the additional property that every point has a neighborhood basis consisting of (absolutely) convex sets. These are called locally convex topological vector spaces, abbreviated as LCVTVS.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary, non-discrete valuated field K; K is endowed with the uniformity derived from its absolute value. The purpose of this generality is to clearly identify those properties of the commonly used real and complex number field that are essential for these basic results. Section 1 discusses the description of vector space topologies in terms of neighborhood bases of 0, and the uniformity associated with such a topology. Section 2 gives some means for constructing new topological vector spaces from given ones. The standard tools used in working with spaces of finite dimension are collected in Section 3, which is followed by a brief discussion of affine subspaces and hyperplanes (Section 4). Section 5 studies the extremely important notion of boundedness. Metrizability is treated in Section 6. This notion, although not overly important for the general theory, deserves special attention for several reasons; among them are its connection with category, its role in applications in analysis, and its role in the history of the subject (cf. Banach [1]). Restricting K to subfields of the complex numbers, Section 7 discusses the transition from real to complex fields and vice versa.

We prove that a mapping from the product of two complete metrizable vector spaces into a topological vector space which is separately universally measurable and separately convex with respect to a convex cone is continuous.0. Introduction.If X is a complete separable vector space and if / is a midpoint convex mapping from X into the real line R which is Christensen measurable, then F. Fischer and Z. Slodkowski have proved in [4] that / is continuous by showing that {(x, r) G X X R: f(x) < r} is open in X X R. Results of this kind about continuity of universally measurable morphisms were obtained before by A. Douady and L. Schwartz (see [7]) for linear operators between locally convex spaces, and by J. P. R. Christensen in [3] for homomorphisms between topological complete metrizable groups.In this paper we are concerned with the study of convex operators taking values in a vector space.Vector analogues of the preceding results are proved.In fact, we estabhsh that a separately universally measurable mapping /: X X Y - F, where X and Y are two complete metrizable vector spaces and F is any topological vector space, is continuous whenever it is separately midpoint convex with respect to a convex cone.Let us also note that many other interesting results about continuity of convex operators are given by J. M. Borwein in [1].

The main objective of this paper is to introduce a new generalized form of topological vector spaces, namely, almost $\\alpha$-topological vector spaces by using the concepts of $\\alpha$-open sets, regular open sets, and almost $\\alpha$-continuous mappings. In the present paper, some important examples of almost $\\alpha$-topological vector spaces with basic properties are characterized.

In I(L)-topological vector spaces (I(L)-tvs) that are L-topologically generated (L-tg), we investigate the generalized local boundedness in this article. We prove that an L-tg I(L)-tvs is generalized locally bounded iff the underlying L-topological vector space(L-tvs) is. We further verify that an L-tg I(L)-tvs is generalized L-fuzzy normable iff the underlying L-tvs is.

A note on multiplier convergent series

In this paper, we introduce the notion of almost \(s-\)topological vector spaces and present some examples and counterexamples of almost \(s-\)topological vector spaces. Some general properties of almost \(s-\)topological vector spaces are investigated.

Application of neustadt's theory of extremals to an optimal control problem with a functional differential equation and a functional inequality constraint

LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D → Y is midconvex,g: D → Y ∪ {− ∞} is midconcave andg ⩽ f onD, then there exists a Jensen mappingh: D → Y ∪ {− ∞} such thatg ⩽ h ⩽ f onD. Using this theorem we show that a mappingf: D → Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D → Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.

Certain results in connection with boundedness and precompactness, which are valid in topological vector spaces and locally bounded prelimit vector spaces, are extended to the class of strictly *-equable pre-limit vector spaces. This class contains the class of topological vector spaces. Hahn-Banach extension and separation theorems hold in the class of strictly *-equable prelimit vector spaces.

Helly and Klee type intersection theorems for finitary connected paved spaces

Our aim is to give other proofs of some slight generalizations of results from Baron (Aequat Math, 2013). We describe larger classes of discontinuous additive involutions $${a:X\to X}$$ on a topological vector space X such that $${a(H)\setminus H\neq\emptyset}$$ holds for a sufficiently numerous set $${H\subset X}$$ of vectors linearly independent over $${{\mathbb{Q}}}$$ . We also consider the topological vector space $${{\mathcal{A}}_X}$$ of all additive functions $${a:X\to X}$$ with the topology induced by the Tychonoff topology of the space X X . We prove in a simple way that some classes of discontinuous additive involutions are dense in the topological vector space $${{\mathcal{A}}_X}$$ .

The mutational equations of Aubin extend ordinary differential equations to metric spaces (with compact balls). In first-order geometric evolutions, however, the topological boundary need not be continuous in the sense of Painleve–Kuratowski. So this paper suggests a generalization of Aubin’s mutational equations that extends classical notions of dynamical systems and functional analysis beyond the traditional border of vector spaces: Distribution– like solutions are introduced in a set just supplied with a countable family of (possibly non-symmetric) distance functions. Moreover their existence is proved by means of Euler approximations and a form of “weak” sequential compactness (although no continuous linear forms are available beyond topological vector spaces). This general framework is applied to a first-order geometric example, i.e. compact subsets of the Euclidean space evolving according to the nonlocal properties of both the current set and its proximal normal cones. Here neither regularity assumptions about the boundaries nor the inclusion principle are required. In particular, we specify sufficient conditions for the uniqueness of these solutions.