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Fragmentability and compactness in C(K)-spaces

100%

Cascales B., Manjabacas G., Vera G.

Studia Mathematica

1998

tom 131

nr 1

73-87

Let K be a compact Hausdorff space, $C_p(K)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p(D)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p(K)$ is Lindelöf the $t_p(D)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p(K)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that $(C(K),t_p(D))$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, $C_p(K)$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex $t_p(D)$-compact subsets of C(K) have the weak Radon-Nikodym property.

Kadec norms and Borel sets in a Banach space

100%

Raja M.

Studia Mathematica

1999

tom 136

nr 1

1-16

We introduce a property for a couple of topologies that allows us to give simple proofs of some classic results about Borel sets in Banach spaces by Edgar, Schachermayer and Talagrand as well as some new results. We characterize the existence of Kadec type renormings in the spirit of the new results for LUR spaces by Moltó, Orihuela and Troyanski.

Ograniczanie wyników

2 Studia Mathematica

1 Cascales B.

1 Manjabacas G.

1 Raja M.

1 Vera G.

1 1999

1 1998

Wyniki wyszukiwania

Fragmentability and compactness in C(K)-spaces

Kadec norms and Borel sets in a Banach space