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Let (E, ℱ) be a weakly compactly generated Frechet space and let ℱ0 be another weaker Hausdorff locally convex topology on E. Let X be an ℱ-bounded compact subset of (E, ℱ0). The ℱ0-closed convex hull of X in E is then ℱ0-compact. We also give a new proof, without using Riemann–Lebesgue-integrable (Birkoff-integrable) functions, with the result that if (E, ∥ · ∥) is any Banach space and ℱ0 is fragmented by ∥ · ∥, then the same result holds. Furthermore, the closure of the convex hull of X in ℱ0-topology and in the original topology of E is the same.

In this paper we show that the Lorentz space Lw, 1(0, ∞) has the weak-star uniform Kadec-Klee property if and only if inft>0 (w(αt)/w(t)) > 1 and supt>0(φ(αt) / φ(t))< 1 for all α ∈ (0, 1), where φ(t) = ∫t0 w(s) ds.

A geometric version of the Vitali integral convergence theorem is established. Parameterized versions of results on uniform absolute continuity in spaces of measures suggested by the convergence theorem are studied.

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Krein-Smulian-Type Theorems

The uniform Kadec-Klee property for the Lorentz spaces Lw,1

The Vitali Integral Convergence Theorem and Uniform Absolute Continuity

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3 results

Krein-Smulian-Type Theorems

The uniform Kadec-Klee property for the Lorentz spaces Lw,1

The Vitali Integral Convergence Theorem and Uniform Absolute Continuity