On the weak and pointwise topologies in function spaces

On the weak and pointwise topologies in function spaces II

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

Let X, Y be compact Hausdorff spaces and B£(X), B*(Y)f 0 < a, |3 < 12 (the first uncountable ordinal), the associated Banach spaces of bounded real-valued Baire functions of classes a and 0. If B*(X) =£ B*AX) (which is the case if a =£ 0 and X is not dispersed), then B*(X) is neither linearly isometric to BQ(Y) nor equivalent to B^(Y) in several other ways. B^(X) is linearly isometric to B%(Y) if and only if X is Baire isomorphic to Y. For 1 < a < ft the maximal ideal space of B£(X) for a nondispersed compact space X As not an F-space. 1. Let I bea compact (more generally, completely regular) Hausdorff space and C(X) the space of continuous real-valued functions on X. Let B0(X) = C(X)9 and inductively define Ba(X) for each ordinal a < Q, (Q denotes the first uncountable ordinal) to be the space of pointwise limits of sequences of functions in U | < a B^(X). Let B*(X) be the space of bounded functions contained in Ba(X). With the pointwise operations Ba(X) and B*(X) are lattice-ordered algebras. With the supremum norm B*(X) is a Banach algebra (see [4, §41]). The Baire sets of X of multiplicative class a, denoted by Za(X)9 are defined to be the zero sets of functions in B*(X). Those of additive class a, denoted by CZa(X), are defined as the complements of sets in Za(X). Finally, those of ambiguous class a, denoted by Aa(X), are the sets which are simultaneously in Za(X) and CZa(X). With the set-theoretic operations of union and intersection, Aa(X) is a Boolean algebra for each a < 12. The sets of exactly ambiguous class a, denoted by EAa(X are those in ^ a ( Z ) U ^ < a Ac(X). The sets of exactly additive and exactly multiplicative class a are defined analogously. The class of all Baire subsets of X is AMS (MOS) subject classifications (1970). Primary 06A65, 26A21, 28A05, 46E1S, 46E30, 46J10, 54C50, 54H05; Secondary 04A15.

On continuous selections for metric projections in spaces of continuous functions

Function-space compactifications of function spaces

0. We consider transition probability operators P(x,E) on a compact metric space which satisfy a rather strong condition (uniform stability in mean). ?1 is devoted largely to notation and terminology. In ?2 an ergodic decomposition is obtained. It is essentially a sharpening of a decomposition of Yosida's valid under less stringent conditions [17]. In ?3 we discuss the behavior of sample paths of the Markov processes themselves relative to the ergodic decomposition obtained in ?2. In ?4 we give some examples of transition probability operators which are uniformly stable in mean. Most of the results of ?2 and some of the results of ?4 appeared in the author's thesis, written under the guidance of Professor Blackwell. Tbe results of ?3 had to await Breiman's discovery of a new version of the strong law of largenumbers. Our Theorem 3.2 is a minor generalization of Breiman's theorem and our proof a slight modification of his. 1. S is a compact metric space. Its metric is denoted by d. , is the a-field generated by the open subsets of S. C(S) is the Banach space of real-valued continuous functions on S with the uniform norm. The dual C*(S) consists of all the finite countably additive measures on S. The value of a member p of C*(S) at a member f of C(S) will be denoted by either (f, ,u) or f f dy. A countably additive measure M on any a-field over any set Q is called a probability measure if it is non-negative and if ,u(Q) = 1. We sometimes refer to probability measures in C*(S) as distributions. A function P on S x I is called a transition probability operator if (i) P(x, * ) is a probability distribution for each x E S, (ii) P( , E) is a s-measurable function for each E e Z. Unless otherwise stated, we will from now on assume that P is a fixed transition probability operator. We will abuse the language by using 'P(x,E)' rather than 'P' to denote not only a value of P but P itself. Associated with P(x,E) are operators T and U, the first defined on bounded X-measurable functions f by (Tf)(x)

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

Given a compact convex planar domain $\Omega $ with non-empty interior, the classical Neumann’s configuration constant $c_{\mathbb{R}}(\Omega )$ is the norm of the Neumann–Poincaré operator $K_\Omega $ acting on the space of continuous real-valued functions on the boundary $\partial \Omega $, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(\Omega )$ of $K_\Omega $ on the corresponding space of complex-valued functions, and the norm $a(\Omega )$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(\Omega ) = c_{\mathbb{C}}(\Omega )$, the analytic Neumann-type inequality $a(\Omega ) &amp;lt; 1$, and provide various estimates for these quantities expressed in terms of the geometry of $\Omega $. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup _{z \in \Omega } |p(z)|$, where $p$ is a polynomial and $\Omega $ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix–Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(\Omega )}$. In the case that $\Omega $ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse.

Chapter 12 N-Compactness and Its Applications

On the universal completion of pointfree function spaces

Let X be a completely regular Hausdorff space and C(X) the space of continuous real-valued functions on X. The epi-topology on C(X) is derived from the Fell topology on the hyperspace of the closed subsets of X x ℝ by identifying each element of C(X) with its epigraph. When X is locally compact, the concept of epiconvergence of nets in C(X) is precisely convergence in the epi-topology. This paper investigates some properties of C(X) in which epiconvergence plays a key role. A typical result characterizes a property on C(X) in terms of the properties on the base space X

The paper shows that a uniform space $X$ is trans-separable if and only if every pointwise bounded uniformly equicontinuous subset of the space of continuous real-valued functions $C_{c}(X) $ equipped with the compact-open topology is metrizable. This extends earlier results of Pfister and Robertson and also applies to show that if $C_{c}(X) $ is angelic then $X$ is trans-separable. The precise relation among DCCC spaces and trans-separable spaces has been also determined.

Phase-isometries between the positive cones of the Banach space of continuous real-valued functions