First Category Function Spaces Under the Topology of Pointwise Convergence

We examine translation-invariant measures on Banach spaces lp, where p ∈ [1,∞]. We construct analogs of the Lebesgue measure on Borel σ-algebras generated by the topology of pointwise convergence (σ-additive, invariant under shifts by arbitrary vectors, regular measures). We show that these measures are not σ-finite. We also study spaces of functions integrable with respect to measures constructed and prove that these spaces are not separable. We consider various dense subspaces in spaces of functions that are integrable with respect to a translation-invariant measure. We specify spaces of continuous functions, which are dense in the functional spaces considered. We discuss Borel σ-algebras corresponding to various topologies in the spaces lp, where p ∈ [1,∞]. For p ∈ [1,∞), we prove the coincidence of Borel σ-algebras corresponding to certain natural topologies in the given spaces of sequences and the Borel σ-algebra corresponding to the topology of pointwise convergence. We also verify that the space l∞ does not possess similar properties.

Let {mathbb {K}} be a non-trivially valued non-Archimedean complete field. Let ell _{infty }({mathbb {N}}, {mathbb {K}}) [ell _c({mathbb {N}}, {mathbb {K}});c_0({mathbb {N}}, {mathbb {K}})] be the space of all sequences in {mathbb {K}} that are bounded [relatively compact; convergent to 0] with the topology of pointwise convergence (i.e. with the topology induced from {mathbb {K}}^{{mathbb {N}}}). Let X be an infinite ultraregular space and let C_p(X,{mathbb {K}}) be the space of all continuous functions from X to {mathbb {K}} endowed with the topology of pointwise convergence. It is easy to see that C_p(X,{mathbb {K}}) is metrizable if and only if X is countable. We show that for any X [with an infinite compact subset] the space C_p(X,{mathbb {K}}) has an infinite-dimensional [closed] metrizable subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). Next we prove that C_p(X,{mathbb {K}}) has a quotient isomorphic to c_0({mathbb {N}}, {mathbb {K}}) if and only if it has a complemented subspace isomorphic to c_0({mathbb {N}}, {mathbb {K}}). It follows that for any extremally disconnected compact space X the space C_p(X,{mathbb {K}}) has no quotient isomorphic to the space c_0({mathbb {N}}, {mathbb {K}}); in particular, for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has no quotient isomorphic c_0({mathbb {N}}, {mathbb {K}}). Finally we investigate the question for which X the space C_p(X,{mathbb {K}}) has an infinite-dimensional metrizable quotient. We show that for any infinite discrete space D the space C_p(beta D, {mathbb {K}}) has an infinite-dimensional metrizable quotient isomorphic to some subspace ell _c^0({mathbb {N}}, {mathbb {K}}) of {mathbb {K}}^{{mathbb {N}}}. If {mathbb {K}} is locally compact then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _{infty }({mathbb {N}}, {mathbb {K}}). If |n1_{{mathbb {K}}}|ne 1 for some nin {mathbb {N}}, then ell _c^0({mathbb {N}}, {mathbb {K}})=ell _c ({mathbb {N}}, {mathbb {K}}). In particular, C_p(beta D, {mathbb {Q}}_q) has a quotient isomorphic to ell _{infty }({mathbb {N}}, {mathbb {Q}}_q) and C_p(beta D, {mathbb {C}}_q) has a quotient isomorphic to ell _c({mathbb {N}}, {mathbb {C}}_q) for any prime number q.

The projective Menger property and an embedding of Sω into function spaces

For a Tychonoff space $X$, we denote by $C_p(X)$ the space of all real-valued continuous functions on $X$ with the topology of pointwise convergence.  We study a functional characterization of the covering property of Hurewicz.

In this paper we present a topology on the space of real-valued functions defined on a functionally Hausdorff space \(X\) that is finer than the topology of pointwise convergence and for which (1) the closure of the set of continuous functions \(\mathcal{C }(X)\) is the set of upper semicontinuous functions on \(X\), and (2) the pointwise convergence of a net in \(\mathcal{C }(X)\) to an upper semicontinuous limit automatically ensures convergence in this finer topology.

We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

The set of continuous maps of a space X to a space Y equipped with the topology of pointwise convergence will be denoted by Cp(X, Y). For Y = R the notation is abbreviated to Cp(X). We will be interested in connections between the topologies of spaces X and Cp(X). These connections have not been studied extensively. However, some progress has been made recently -interesting results have been obtained in problems of normality and final compactness of Cp(X) (e.g., see [i, 2]). We will study some problems associated with cardinalvalued invariants. All spaces considered are assumed to be completely regular. I. Separability Problems. The problem of separability of Cp(X) was first considered in [3~. The following result was obtained: TIIEOREM i. Cp(X) is separable if and only if X is condensed to a space of countable weight. The main result of the present note is: THEOREM 2. Cp(X) is hereditarily separable if and only if xn is hereditarily finally compact, n~-Ni==_ Proof. Necessity. Let Cp(X) be hereditarily separable. Statement i. Suppose that ~C~Cp(X) is hereditarily separable and separates points from closed sets (i.e., if x~F and F is closed, then there is f ~ ~ such that f(x) = 1 and f(F) ~ 0). Then X is hereditarily finally compact~ Indeed, let ? = {H} be a family of sets open in Xo For each point x ~ (here, ~ = [J {H: H~---_?} is the body of y), we choose H~:? and ]~__~ such that fx(x) = 1 and f~(x \ H~)~0 There exists a countable set B~=-~ such that {/2: x=_B} is dense in {/~: x~}. Then %% = U {H~: x~-B} Indeed, if x:~_~, then there exists g~B subject to condition [/x (x) --f~ (x) [ <: 1/2. Then fy (x) ~ 0 and x ~ I f v.

All spaces considered in this paper are assumed to be metrizable. A compactum is a compact space. A continuum is a connected compactum, and a mapping is a continuous function. For a space X we denote by C(X) the space of all real-valued mappings on X with the topology of uniform convergence. Then by Milutin's interesting work [8], we have known that for each pair of uncountable compacta X and Y, C{X) is linearly isomorphic to C(Y) (see [12] for the detailsand generalizations). On the other hand, for space X we denote by CP(X) the space of all real-valued mappings on X with the topology of pointwise convergence. Spaces X and Y are said to be l-equivalent[1] provided that CP(X) is linearly isomorphic to CP(Y), written CP(X) = CP(Y). Recently, Pavlovskii [11] showed the following.

This paper deals with families of conformal iterated function systems (CIFS). The space of all CIFS, with common seed space X and alphabet I , is successively endowed with the topology of pointwise convergence and a new, weaker topology called $\lambda$ -topology. It is proved that the pressure and the Hausdorff dimension of the limit set are continuous with respect to the topology of pointwise convergence when I is finite, and are lower semi-continuous, though generally not continuous, when I is infinite. It is then shown that these two functions are, in any case, continuous in the $\lambda$ -topology. The concepts of analytic, regularly analytic and plane-analytic families of CIFS are also introduced. It is established that if a family of CIFS is regularly analytic, then the Hausdorff dimension function is real-analytic; if a family is plane-analytic, then the Hausdorff dimension function is continuous and subharmonic, though not necessarily real-analytic. These results are then applied to finite parabolic CIFS. Counter-examples highlighting breakdowns of real-analyticity in the Hausdorff dimension among analytic, but not regularly analytic, families are further provided. Such families often exhibit a phenomenon known as phase transition. Sufficient conditions preventing the occurrence of such transitions are supplemented.

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂cQ, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: l2 → C(X) from a separable Hilbert space l2, where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space Lc(l2, C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.

In this paper we use the following abbreviations: FB (filter base), MS (measurable space), AS (attraction set), DS (directed set), AD (attainability domain), GE (generalized element), s/s (subset), ApS (approximate solution), TS (topological space), and u/f (ultrafilter). We consider general questions connected with the asymptotic attainability; the latter is understood as the attainability of certain states under asymptotic constraints. One natural concrete variant of the problem which we consider below is related to the construction of the AD of a controlled system at a fixed time moment and the study of its properties. This concrete problem is not stable with respect to disturbances, in particular, for weakened constraints. One can treat weakened constraints (phase constraints, edge and intermediate ones) in this problem as asymptotic restrictions. Respectively, the limit of real AD which correspond to weakened constraints can be considered as an AS; from the practical point of view it plays the same role as a “regular ” AD does. Together with AD, with the strict and approximate fulfillment of constraints one can consider sheaves of trajectories of the controlled system and thus obtain sets in a functional space equipped with the corresponding topology. The latter can be generated by a metric, but this is not necessarily so (for example, in cases typical for control problems with continuous time, the topology of pointwise convergence is not metrizable). Therefore it is natural to consider logically similar problems on the attainability in TS, though this can require the application of more complex constructions for AS. This approach is used in the present paper. In addition, using such general statements of the problem, it is natural to assume asymptotic analogs of constraints which are not necessary connected with the weakening of some standard conditions. Really, weakening conditions of such a kind, we form a family of sets of feasible elements for each constraint weakened in a concrete way (generated by the set of mentioned conditions). From this moment we “deal” with the obtained family (really, this is a result of the relaxation procedure for the initial problem), using the minimal set of properties of sets that compose it. But this means that from the beginning we can assume that the family of s/s of the space of regular solutions (or controls) is given, not questioning how this family has been obtained (it is advisable to reduce the set of properties “to the minimum”). In particular, we do not tend to connect this family with a successive weakening of some “rigid ” conditions. In this case the question on the attainability in TS has an asymptotic character. Let us consider a very simple example of a static problem, fixing two finite-dimensional spaces R m and R n (m and n are natural numbers), and a bounded mapping f from R m to R n .F or de finiteness,

The purposes of this note are to complement the results of [4] by citing examples of spectral operators (in the sense of that paper) on non-normable locally convex spaces and to announce a result on the structure of spectral operators which indicates that these examples exhibit rather typical behavior for spectral operators on a large class of locally convex spaces of interest to analysts. We retain the notions of [4] and refer to it for proofs. Let E be a locally convex space (always assumed Hausdorff), £(E) the algebra of continuous linear transformations of E into itself (the identity being denoted by e), £S(E) the algebra £(E) under the topology of pointwise convergence, and let £ if R(-) is holomorphic in a neighborhood of oo. The spectrum of u (denoted by of (&(X) into £8(E); conversely, any continuous homomorphism of 6(X) into £a(E) has associated with it a spectral measure jut (and thus an extension to ($>(X)). The connection between the measure and the homomorphism is given by

The latter defines the usual equivalence relation in Moo. We denote by f the equivalence class of each feM ~ with respect to this equivalence relation. We say that a set F carries p ifF~ and p(E-F)=O. In the case when E is a compact space and/~ a positive Radon measure on E, we denote by Supp ~ the smallest (= intersection) of all closed sets F c E, carrying p. If f: E~R and AcE, we denote by fla the restriction of f to A; finally if HcMoo we use the notation Hla for the set of all hlA, with h~H. The author is indebted to Professors Dietrich K61zow and John C. Oxtoby for several helpful remarks and relevant comments concerning the contents of this paper. We begin with the following simple but very useful result. Proposition 1. Let H c M ~ be a set which is bounded and such that the relations h 1 ~H, h 2 ~H and h I 4: h 2 imply h 1 + h 2 . Assume that H is sequentially compact for the topology of pointwise convergence on E, that is for any sequence (h,) of elements of H, there is a subsequence (h,k) and an h~H such that h,k ~ h pointwise on E. Then H is compact and metrizable for the topology of pointwise convergence on E. Proof We consider on H the topology -[-1 of mean Ll-convergence 1, and the topology T of pointwise convergence on E. We divide the proof into two steps: I) We note first that (H, T1) is compact. In fact, for any sequence (h,) in H, there is a subsequence (h,~) and an h~H such that h,~ ~ h pointwise on E. By the Lebesgue dominated convergence theorem, we also have

Let X be a real Banach space. A subset B of the dual unit sphere of X is said to be a boundary for X, if every element of X attains its norm on some functional in B. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of X which is compact in the topology of pointwise convergence on B is already weakly compact. This problem was recently solved by Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.

Let $\K$ be a complete ultrametric algebraically closed field and let $A$ be the Banach $\K$-algebra of bounded analytic functions in the ''open'' unit disk $D$ of $\K$ provided with the Gauss norm. Let $Mult(A,\Vert \ . \\Vert)$ be the set of continuous multiplicative semi-norms of $A$ provided with the topology of pointwise convergence, let $Mult_m(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult(A,\Vert \ . \ \Vert)$ whose kernel is a maximal ideal and let $Mult_1(A,\Vert \ . \ \Vert)$ be the subset of the $\phi \in Mult(A,\Vert \ . \ \Vert)$ whose kernel is a maximal ideal of the form $(x-a)A$ with $ a\in D$. By analogy with the Archimedean context, one usually calls ultrametric Corona problem, or ultrametric Kakutani problem the question whether $Mult_1(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$. In a previous paper, we have recalled the characterization of a large set of continuous multiplicative semi-norms and why the multbijectivity of the algebra $A$ would solve the Corona problem. Here we prove that multbijectivity in the general case. This implies that $Mult_1(A,\Vert \ . \ \Vert)$ is dense in $Mult_m(A,\Vert \ . \ \Vert)$, beginning by the case when $\K$ is spherically complete and generalizing next.