Chapter 12 N-Compactness and Its Applications

Chapter 52 - The interplay between compact spaces and the Banach spaces of their continuous functions

In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-Cech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrowka in [4]; second, the idea included in the proof of the theorem: “Every compact Baire set is aG δ ” as given inHalmos' text on measure theory [2; Section 51, theorem D].

A n-dimensional topological manifold is defined as an n-dimensional local Euclidean space M that is also a second countable space and a Hausdorff space. If a topological space has a countable base, it is referred to as a second countable space. At the same time, continuous function and metric space can help people to understand topological space better. This article will express some basic ideas about continuous functions, metric space, topological space and its properties, and topological manifolds. Moreover, the paper shows some basic ideas of how topological manifolds, topological space and function could be recognized and proved. Through analysis, this paper demonstrates the connection between them, such as using properties of continuous function to prove the definition and properties of topological manifolds and spaces.

For a compact space K we denote by $$C_w(K)$$ ( $$C_p(K)$$ ) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose that K is an infinite (metrizable) compact space. Can $$C_w(K)$$ and $$C_p(K)$$ be homeomorphic? We show that the answer is “no”, provided K is an infinite compact metrizable C-space. In particular our proof works for any infinite compact metrizable finite-dimensional space K.

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.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

We prove that if K is a compact space and the space P(K K) of regular probability measures on K K has countable tightness in its weak topology, then L1( ) is separable for every 2 P(K). It has been known that such a result is a consequence of Martin's axiom MA(!1). Our theorem has several consequences; in particular, it general- izes a theorem due to Bourgain and Todor£evi¢ on measures on Rosenthal compacta. 1. Introduction. The tightness of a topological space X, denoted by (X), is the least cardinal number such that for every A X and x2 A there is a set A0 A withjA0j (X) and such that x2 A0. Throughout, K stands for a compact Hausdor topological space. By

In a recent paper, F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E), the Banach space of continuous functions from a compact Hausdorff space Q to E, has the Dieudonne property. They asked whether or not the result is still true if one only assumes that E does not contain a copy of l1. In this paper we give a positive answer to their question. As a corollary we show that if E is a subspace of an order continuous Banach lattice, then E has the Dieudonne property if and only if C(Q, E) has the same property. If E is a Banach space and Q is a compact Hausdorff space, then C(Q, E) will stand for the Banach space of the E-valued continuous functions on Q under the supremum norm. A Banach space E is said to have the Dieudonne property if for every Banach space F, any bounded linear operator T: E -> F that transforms weakly Cauchy sequences into weakly convergent sequences is weakly compact. In [3] F. Bombal and P. Cembranos showed that if E is a Banach space such that E* is separable, then C(Q, E) has the Dieudonne property and they asked whether the same result is true when replacing the assumption that E* is separable by supposing only that 11 does not embed in E. In this paper we give a positive answer to their question. Recall that a topological space (X, -y) is said to be Polish if it is homeomorphic to a separable complete metric space and it is said to be analytic if it is the continuous image of a Polish space. A subset A of a topological space (X, y) is said to be coanalytic if its complement (X\A, y) is analytic. Finally A is said to be PCA if it is the continuous image of a coanalytic space. The notations and terminology used and not defined can be found in [5, 8, or 10]. In the proof of Lemma 3 we need the following two results. THEOREM 1 (M. SREBRNY [9]). Let X and Y be two analytic spaces and let F be a multivalued function from X to the subsets of Y, such that its graph is PCA and for which one can prove that for every x c X, F(x) :8 0 using only the axioms of ZFC. Then there exists a universally measurable map f: X -> Y such that f (x) c F(x) for every x E X. THEOREM 2 (I. ASSANI [1, 2]). Let E be a separable Banach space. The set of weakly Cauchy sequences is a coanalytic subset of EN. Received by the editors January 5, 1985. 1980 Mathematics Subject Classification. Primary 46G10, 46B22.

Almost compactness and near compactness in fuzzy topological spaces

UDC 515.122 We consider the problem of characterization of topological spaces embedded into countably compact Hausdorff topological spaces. We study the separation axioms for subspaces of Hausdorff countably compact topological spaces and construct an example of a regular separable scattered topological space that cannot be embedded into a Urysohn countably compact topological space.

αo̧mpactness in bifuzzy topological spaces

On the weak and pointwise topologies in function spaces II

A classical result due to M. Eidelheit and B. Yood states that the standard algebra norm on the algebra of bounded linear operators on a Banach space is minimal, in the sense that the norm must be less than a multiple of any other submultiplicative norm on the same algebra. This definition does not assume that the arbitrary algebra norm is complete. In cases when the standard algebra norm is, in addition, maximal, it is therefore unique up to equivalence. More recently, M. Meyer showed that the Calkin algebras of a very restricted class of Banach spaces also have unique algebra norms. We generalise the Eidelheit-Yood method of proof, to show that the conventional quotient norm on a larger class of Calkin algebras is minimal. Since maximality of the norm is a presumed property for the class, the norm is also unique. We thus extend the result of Meyer. In particular, we establish that the Calkin algebras of canonical Banach spaces such as James’ space and Tsirelson’s space have unique algebra norms, without assuming completeness. We also prove uniqueness of norm for quotients of the algebras of operators on classical non-separable spaces, the closed ideals of which were previously studied by M. Daws. One aspect of the Eidelheit-Yood method is a dependence on the uniform boundedness principle. As a component of our generalisation, we prove an analogue of that principle which applies to Calkin algebra elements rather than bounded linear operators. In order to translate the uniform boundedness principle into this new setting, we take the perspective that non-compact operators map certain wellseparated sequences to other well-separated sequences. We analyse the limiting separation of such sequences, using these values to measure the non-compactness of operators and define the requisite notion of a bounded set of non-compact operators. In the cases when the underlying Banach space has a Schauder basis, we are able to restrict attention to seminormalised block basic sequences. As a consequence, our main uniqueness of norm result for Calkin algebras relies on the existence of bounded mappings between, and projections onto, the spans of block basic sequences in the relevant Banach spaces.

It is proved that if -V is a metric space, E a Banach space containing a a-weakly-compact dense subset, then the space (Mr(X, E'), o(MT(X, £"), Cb(X, E))) is angehe, Cb(X, E) being all bounded continuous functions from X into E and MT(X, £") the dual of Cb(X, E) with the strict topology ß.A Hausdorff topological space Y is called angelic if (i) every relatively countably compact subset of Y is relatively compact, and (ii) for any point x in the closure of a relatively compact subset A of Y, there exists a sequence, in A, converging to x [4, p. 534].In this paper X will stand for a completely regular Hausdorff space, E a Banach space over K, the field of real or complex numbers, C(X, E) (C(X)) all £-valued (AT-valued) continuous functions on X, and Cb(X, E) (Cb(X)) all bounded £-valued (/T-valued) continuous functions on X.We shall use the notations of [5] for locally convex spaces.Also the notations and results from [2] will be used.The topologies ß, ß0, ßx, ße (also denoted by ßx) are defined on Cb(X) in [9], [7],[8]-these topologies are defined for K = R, the reals, but naturally extend to K = C, the complex field, ß, ß0, ßx, ßx are defined on Cb(X, E) in [1], [2].If %°° = %X(X, E) = [H c Cb(X, E): H pointwise equicontinuous and uniformly bounded}, the topology ßx is the finest locally convex topology on Cb(X, E) agreeing with pointwise topology on each H E %°°.It is well known that MT(X, E') = (Cb(X, E), ß)', M,(X, E') = (Cb(X, E), ß0)' and MX(X, E') = (Cb(X, E), ßj (see [2]).If J is a metric, MX(X, E') = Mr(X, £") and ß and ßx are both Mackey [2]; from this it follows that ß = ßx in this case.If X is a metric space it is proved in [5, Theorem 5.3] that (Moe(X), o(Mao(X), Cb(X))) is angelic.We will extend this result to MX(X, E') for some special Banach spaces E.Theorem 1. (F', t(F', F)) is complete, t denoting the Mackey topology of the duality <F, F'), where F = Cb(X, E) and F' = MX(X, E').Proof.By Grothendieck completeness theorem [5, Theorem 6.2, p. 148], it is enough to prove that any linear form p on F which is continuous on every

The main aim of this paper is to study new class of continuous function is called – continuous function. for this aim, the nation of - open and pre – open sets and - compact space are introduced. and we shall study the relationship between – continuous and pc– continuous functions.