Let K be an arbitrary compact space and C(K) the space of continuous functions on K endowed with its natural supremum norm. We show that for any subset B of the unit sphere of C(K)* on which every function of C(K) attains its norm, a bounded subset A of C(K) is weakly compact if, and only if, it is compact for the topology tp(B) of pointwise convergence on B. It is also shown that this result can be extended to a large class of Banach spaces, which contains, for instance, all uniform algebras. Moreover we prove that the space (C(K), tp(B)) is an angelic space in the sense of D. H. Fremlin.

Given a topological space X, we denote by Cp(X) the space of real-valued continuous functions on X, equipped with the topology of pointwise convergence.

We prove a characterization of functions in B1/4(K)\C(K), where K is a compact metric space in terms of c0-spreading models, answering a Problem of R. Haydon, E. Odell and H. Rosenthal. Beginning with B1/4(K) we define a decreasing family (Vξ(K),║ · ║ξ)1≤ξ<ω1 of Banach spaces whose intersection is DBSC(K) and we prove an analogous stronger property for the functions in Vξ(K)\C(K). Defining the s-spreading model-index, we classify B1/4;(K) and we prove that s-SM[F]>ξ for every F∈ Vξ(K). Also we classify the separable Banach spaces by defining the c0-SM-index which measures the degree to which they have sequences with extending spreading models equivalent to the usual basis of c0. We give examples of Baire-1 functions and reflexive spaces with arbitrary large indices.

Let X be a completely regular space, and Cb(X) the space of all bounded continuous real valued functions on X equipped with the metric associated to the uniform norm. For f∈Cb(X) and γ∈¡ we use the following standard notations: inf(f) = infx∈Xf(x) and {f<γ} = {x∈X:f(x)<γ}

In the part (16-3) of his extensive study on measurability in Banach spaces, Talagrand [12] considered the Banach space C(K) of continuous functions on a dyadic topological space K. He proved that C(K) is realcompact in its weak topology, if, and only if, the topological weight of K is not a twomeasurable cardinal (Theorem 16-3-1). Then he asked for an alternative to a rather complicated proof presented there (p. 214) and posed the problem whether C(K) is measure-compact whenever the weight of K is not a realmeasurable cardinal (Problem 16-3-2).

In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

Le point de départ de ce travail est le résultat suivant.

THÉORÈME (I. Namioka). Soient X et Y deux espaces compacts et

Si f est continue quand on munit(Y) de la topologie de convergence simple alors X contient un Gδ dense en tout point duquel f reste continue quand on munit(Y) de la topologie de la convergence uniform.

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation