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The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces

S. Gabriyelyan, J. Kąkol, G. Plebanek (2016)

Studia Mathematica

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of C k ( X ) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k ℝ -space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C k ( X ) is Ascoli iff C k ( X ) is a k ℝ -space iff X is locally compact. Moreover, C k ( X ) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...

The associated tensor norm to ( q , p ) -absolutely summing operators on C ( K ) -spaces

J. A. López Molina, Enrique A. Sánchez-Pérez (1997)

Czechoslovak Mathematical Journal

We give an explicit description of a tensor norm equivalent on C ( K ) ⊗ F to the associated tensor norm ν q p to the ideal of ( q , p ) -absolutely summing operators. As a consequence, we describe a tensor norm on the class of Banach spaces which is equivalent to the left projective tensor norm associated to ν q p .

The Banach algebra of continuous bounded functions with separable support

M. R. Koushesh (2012)

Studia Mathematica

We prove a commutative Gelfand-Naimark type theorem, by showing that the set C s ( X ) of continuous bounded (real or complex valued) functions with separable support on a locally separable metrizable space X (provided with the supremum norm) is a Banach algebra, isometrically isomorphic to C₀(Y) for some unique (up to homeomorphism) locally compact Hausdorff space Y. The space Y, which we explicitly construct as a subspace of the Stone-Čech compactification of X, is countably compact, and if X is non-separable,...

The Banach space S is complementably minimal and subsequentially prime

G. Androulakis, T. Schlumprecht (2003)

Studia Mathematica

We first include a result of the second author showing that the Banach space S is complementably minimal. We then show that every block sequence of the unit vector basis of S has a subsequence which spans a space isomorphic to its square. By the Pełczyński decomposition method it follows that every basic sequence in S which spans a space complemented in S has a subsequence which spans a space isomorphic to S (i.e. S is a subsequentially prime space).

The Banach-Saks property and Haar null sets

Eva Matoušková (1998)

Commentationes Mathematicae Universitatis Carolinae

A characterization of Haar null sets in the sense of Christensen is given. Using it, we show that if the dual of a Banach space X has the Banach-Saks property, then closed and convex subsets of X with empty interior are Haar null.

The Banach-Saks property in rearrangement invariant spaces

P. G. Dodds, E. M. Semenov, F. A. Sukochev (2004)

Studia Mathematica

This paper studies the Banach-Saks property in rearrangement invariant spaces on the positive half-line. A principal result of the paper shows that a separable rearrangement invariant space E with the Fatou property has the Banach-Saks property if and only if E has the Banach-Saks property for disjointly supported sequences. We show further that for Orlicz and Lorentz spaces, the Banach-Saks property is equivalent to separability although the separable parts of some Marcinkiewicz spaces fail the...

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