On the support of midconvex operators

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.

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.

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

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.

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.

Internal points were introduced in the literature of topological vector spaces to characterize the finest locally convex vector topology. Later on, they have been generalized to the context of real vector spaces by means of the inner points. Inner points can be seen as the most opposite concept to the extreme points. In this manuscript we solve the “faceless problem”, that is, the informally posed open problem of characterizing those convex sets free of proper faces. Indeed, we prove that a non-singleton convex subset of a real vector space is free of proper faces if and only if every point of the convex set is an inner point. As a consequence, we prove the following dichotomy theorem: a point of a convex set of a real vector space is either a extreme point or an inner point of some face. We also characterize the non-singleton closed convex subsets of a topological vector space free of proper faces, which turn out to be the linear manifolds. An application of this allows us to show that the only possible minimal faces of a linearly bounded closed convex subset of a topological vector space are the extreme points. As an application of the technique we develop to prove our results, we easily construct dense proper faces of convex sets and non-dense proper faces whose closure is not a face. This fact allows us to equivalently renorm any infinite dimensional Banach space so that its unit ball contains a non-closed face. Even more, we prove that every infinite dimensional Banach space containing an isomorphic copy of $$c_0$$ or $$\ell _p$$ , $$1<p<\infty $$ , can be equivalently renormed so that its unit ball contains a face whose closure is not a face.

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 functor KR(X), defined in [4], is a contravariant functor defined in the category of spaces with involutions.It is shown herein that this functor is classified by equivariant maps into the complex Grassmann manifold, which is given the involution induced by complex conjugation.For the case of free involutions it is shown that the classifying maps can be taken to lie outside the fixed point set of the Grassmann manifold.This fixed point set can be identified with the real Grassmann manifold.It is then shown that, for free involutions, KR(X) is an invariant of the homotopy type of the orbit space X modulo its involution.The multiplicative group of real line bundles, real in the sense of [4], is shown to be classified by equivariant maps into a quadric surface Q in complex projective space.Q carries a free involution and this classification is again valid for spaces with free involutions. I. Introduction.In [4] the definition of Atiyah-real vector bundles was given, and they, together with the associated A-theory, were studied.The purpose of this paper is twofold.First we will extend the known classification theorems for complex A-theory to the new theory.Second we will obtain new results for the case of Atiyah-real vector bundles over spaces with free involutions.It was anticipated that this functor could be used to obtain a characterization of Mk, the differentiable seven sphere defined by Milnor in [1].This question is studied in 111, where KR{Ml) is computed.To review the definitions and results of [4], we have that a real space is a topological space X together with a homeomorphism t: X -> X with t2 = identity.We will write ?(x) = x.An Atiyah-real vector bundle over a real space is a complex vector bundle E -> X which itself is a real space and such that (1) 7r: 7i-> Xis equivariant, (2) the induced map Ex -> Es is conjugate linear.The Grothendieck group of Atiyah-real vector bundles over the real space X is denoted by KR(X).There is a Bott periodicity theorem and derived cohomology theory.Define(1) Rp-" = R'i + iRp,

Let K be a compact Hausdorff space and let )~ be a probability measure on K. We denote by Lo(K,2) the space of all Borel functions f:K--*N with the topology of convergence in measure. Lo(K, 2) is an F-space (complete metric topological vector space) if, as usual, we identify functions equal almost everywhere. Spaces of the type Lo(K,2 ) are probably the most studied examples of non-locally convex topological vector spaces. In spite of their bad reputation there is some evidence that they are in fact rather well-behaved spaces. Thus for a general topological vector space X it is often very useful to be able to produce nontrivial linear operators T:X--*Lo(K,2 ), in the same way as linear functionals facilitate the theory of locally convex spaces. We shall say that a point xeX is pathological it wherever T:X-*Lo(K, 2) is a continuous linear operator then Tx=O. X is pathological if every xsX is pathological. Note that if X is separable then it suffices in the definition to take K =(0, 1) with Lebesgue measure. The first example of a pathological F-space was given in 1973 by Christensen and Herer [3]. A more natural example is the space Lp/Hp where 0 < p < 1 (see [1, 5]). However the Christensen-Herer example also showed the connection with pathological submeasures. We recall that if d is an algebra of subsets of some abstract set L then a submeasure 4~:d--,lR is a map satisfying ~b(A)<_~b(AwB)<d~(A)+O(B) and th(O) = 0. 4~ is said to be pathological (see [3, 11]) if whenever 2 : d ~ N is a (finitelyadditive) measure with 0 <)~(A) < th(A) (At d ) then )~0. Now if S(~) denotes the space of all simple d-measurable functions f:L~IR, then S(~) can be topologized by the topology of convergence in @measure (functions differing only on a set of @measure zero are identified). The completion of S(~) in this topology may be denoted by A(~b). Then A(q~) is pathological

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.

The various properties of topological spaces in the classical Baire category theorems which imply the Baire property also imply the stronger property of pseudo-completeness. In contrast to some of these properties and to the Baire property, J. C. Oxtoby has shown that pseudo-completeness is productive. The following main result places pseudo-completeness in the context of linear topological spaces: Let E and F be linear topological spaces and g be a continuous linear mapping of E into F. If E is pseudo-complete, g is almost open, and the completion of g[E] has a continuous metric, then g[E] is complete. The proof of this result uses the difference theorem, but not an open mapping theorem. The hypotheses lead to a discussion of conditions for a linear mapping to be almost open and for a linear topological space to have a continuous metric. An example shows that, although a translation invariant continuous metric on a linear topological space E extends to a translation invariant continuous pseudo-metric on the completion of E, this extension need not be a metric, even if it induces a normed topology on E. Othes examples show that a pseudo-complet e linear topological space need not be complete in its natural uniformity, and that the almost open condition of the main result may not be omitted and is not implied by the combination of the other conditions and the conclusion. Pseudo-complete topological spaces, introduced by Oxtoby [8], successfully bind together the classical Baire category theorems. They have been the subject of recent work; Aarts and Lutzer [1], for example, have shown that a metrizable space is pseudo-comple te if and only if it has a dense subspace which is topologically complete. In the context of linear topological spaces, Todd [13] shows that, under certain conditions, a linear topological space which is pseudocomplete is complete in the natural uniformity. Theorem 3.1 extends this result for the image of a pseudo-complete linear topological space under a continuous linear almost open mapping; in contrast. Example 5.5 gives pseudo-complete locally convex linear topological spaces which are not complete, and hence are not Ptak spaces. From open mapping theorems arise restrictions on continuous images of certain types of spaces. As an application of his open mapping theorem, Banach ([2], p. 38), showed that the continuous linear image of a complete metric linear space in a space of the same

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.

In this paper, we have introduced a new generalized form of topological vector spaces, namely, almost β-topological vector spaces by using the concept of β-open sets. We have also presented some examples and counterexamples of almost β-topological vector spaces and determined its relationship with topological vector spaces. Some properties of β-topological vector spaces are also characterized.

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.