81 research outputs found
An extension of Lorentz's almost convergence and applications in Banach spaces
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.
(b) Tn α, n ≥ 0 admits an l1-subsequence and a nontrivial weakly Cauchy subsequence while a is almost convergent.
Finally we show that, in the sense of measure, for almost all real sequences taking values in a compact set K ⊆ R (with at least two points), the sequence (Tn α)n ≥ 0 is equivalent in the supremum norm to the usual l1-basis and (hence) not almost convergent
Compactness in the First Baire Class and Baire-1 Operators
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, E) is relatively compact, etc. We also show that our class includes
Gulko compact.
In the second part of the paper we examine under which conditions a
bounded linear operator T : X ∗ → Y so that T |BX ∗ : (BX ∗ , w∗ ) → Y is a
Baire-1 function, is a pointwise limit of a sequence (Tn ) of operators with
T |BX ∗ : (BX ∗ , w∗ ) → (Y, · ) continuous for all n ∈ N. Our results in this
case are connected with classical results of Choquet, Odell and Rosenthal
Sequential closure in the space of measures
We show that there is a compact topological space carrying a measure which is
not a weak* limit of finitely supported measures but is in the sequential
closure of the set of such measures. We construct compact spaces with measures
of arbitrarily high levels of complexity in this sequential hierarchy. It
follows that there is a compact space in which the sequential closure cannot be
obtained in countably many steps. However, we show that this is not the case
for our spaces where the sequential closure is always obtained in countably
many steps.Comment: (18 pages, a gap in an argument from the previous version fixed
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