For a polish space M and a Banach space E let B1 (M, E)
be the space of first Baire class functions from M to E, endowed with the
pointwise weak topology. We study the compact subsets of B1 (M, E) and
show that the fundamental results proved by Rosenthal, Bourgain, Fremlin,
Talagrand and Godefroy, in case E = R, also hold true in the general
case. For instance: a subset of B1 (M, E) is compact iff it is sequentially
(resp. countably) compact, the convex hull of a compact bounded subset of
B1 (M,...
2000 Mathematics Subject Classification: Primary 40C99, 46B99.
We investigate an extension of the almost convergence of G. G. Lorentz requiring that the means of a bounded sequence converge uniformly on a subset M of N. We also present examples of sequences α∈ l∞(N) whose sequences of translates (Tn α)n≥ 0 (where T is the left-shift operator on l∞(N)) satisfy:
(a) Tn α, n ≥ 0 generates a subspace E(α) of l∞(N) that is isomorphically embedded into c0 while α is not almost convergent.
...
A class of Banach spaces, countably determined in their weak topology (hence, WCD spaces) is defined and studied; we call them strongly weakly countably determined (SWCD) Banach spaces. The main results are the following: (i) A separable Banach space not containing ℓ¹(ℕ) is SWCD if and only if it has separable dual; thus in particular, not every separable Banach space is SWCD. (ii) If K is a compact space, then the space C(K) is SWCD if and only if K is countable.
Download Results (CSV)