Comment on Strongly Preirresolute Topological Vector Spaces

Helly and Klee type intersection theorems for finitary connected paved spaces

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.

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.

In this paper, we introduce sufficient conditions for the non—empty intersection of two set—valued mappings in topological spaces. As applications, some topological minimax inequalities for two functions in which one of them is separately lower (or upper) semicontinuous are given. Finally, by employing our topological intersection theorems for two set—valued mappings, some other minimax inequalities have been derived without separately lower (or upper) sernicontinuity under but with another condition. These results are topological versions of corresponding minimax inequalities for two functions due to Fan (1964) and Sion (1958) in topological vector 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 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 this paper, we first introduce the new notions of $$\overline{{\mathbb {R}}}_+$$-local-intersection property, $$\overline{{\mathbb {R}}}_+$$-local-inclusion property, $$\overline{{\mathbb {R}}}_+$$-strongly transfer lower semicontinuity, $${\mathbb {R}}_-$$-weakly transfer lower semicontinuity and $${\mathbb {R}}_-$$-strongly transfer lower semicontinuity for a set-valued mapping $$\Phi $$ in topological vector spaces. Then, using these notions, we characterize the existence of set-valued equilibrium without assuming any form of convexity of function and/or convexity and compactness of sets. Furthermore, we apply the basic results obtained in the paper to Browder variational inclusion, with weekend conditions on the involved set-valued operators and provide an affirmative answer to some open question posed by Alleche and Rǎdulescu in their final remark of (Optim Lett, 2018. https://doi.org/10.1007/s11590-1233-2). Some application of our results to characterize the existence of set-valued pure strategy Nash equilibrium in games with discontinuous and non-convex payoff functions and nonconvex and/or noncompact strategy spaces is presented. Finally, we characterize the existence of single-valued equilibrium problem without assuming any form of convexity of function and/or convexity and compactness of sets in the setting of topological vector spaces. Our results improve and generalize many known results in the current literature.

Incomplete infima of complete uniformities related to topological groups and topological vector spaces

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 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.

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.

It is proved that the free topological vector space $\mathbb{V}([0,1])$ contains an isomorphic copy of the free topological vector space $\mathbb{V}([0,1]^{n})$ for every finite-dimensional cube $[0,1]^{n}$, thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval $[0,1]$ to general metrisable spaces. Indeed, we prove that the free topological vector space $\mathbb{V}(X)$ does not even have a vector subspace isomorphic as a topological vector space to $\mathbb{V}(X\oplus X)$, where $X$ is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line.

In [1], we proved a criterion for the disjointness of two induced representations UL and UM of a Lie group G, where L and M are finite-dimensional unitary representations of compact subgroups H and K, respectively, of G. The purpose of this paper is to improve this theorem by getting a stronger conclusion, while dropping the conditions that G be a Lie group arld H be compact, and that L and M be finite-dimensional. Moreover, the restriction on K is weakened to read: G has arbitrarily small neighborhoods of the identity invariant under the adjoint action of K on G. Finally, the proof given below is fairly elementary, while the proof in [I ] is quite involved. Notations and conventions: Let Al be a topological vector space. C(G, CU) will denote the space of continuous functions from G to cA, equipped with the topology of uniform convergence on compact subsets of G while Co(G, CLL) will denote the space of those fC-C(G, CLL) with compact support. If cAl is omitted it is understood that ca=C. If cUl is another topological vector space, ?(ca, cU4) will denote the space of continuous linear maps from cA into cal equipped with the topology of bounded convergence. All integrations are with respect to right Haar measure. For any locally compact group G, G will denote its modular function. If f, g Co (G), f o g will denote the convolution of f and g, and f* is defined by f *(x) = c G(x) -f(x-1)If L and M are representations of G, R(L, M) will denote the space of intertwining operators for L and M (see [3]), while I(L, M) will denote the dimension of R(L, M). For the definition of induced representation used below, see [1]. Finally, for any function f on the group G and any xGG, fx and fx are defined by fx(y) =f(x-ly) and fx(y) =f(yx).

We present a theory of measure and integration in topological vector spaces and generalize the Fichtenholz-Kantorovich-Hildebrandt and Riesz representation theorems to this setting, using strong integrals.As an application, we find the containing Banach space of the space of continuous p-normed space-valued functions.It is known that Bochner integration in p-normed spaces, using Lebesgue measure, is not well behaved and several authors have developed integration theories for restricted classes of functions.We find conditions under which scalar measures do give well-behaved vector integrals and give a method for constructing examples.