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Strong topologies on vector-valued function spaces

Marian Nowak (2000)

Czechoslovak Mathematical Journal

Let ( X , ∥ · ∥ X ) be a real Banach space and let E be an ideal of L 0 over a σ -finite measure space ( Ø , Σ , μ ) . Let ( X ) be the space of all strongly Σ -measurable functions f Ø → X such that the scalar function f ˜ , defined by f ˜ ( ø ) = ∥ f ( ø ) ∥ X for ø ∈ Ø , belongs to E . The paper deals with strong topologies on E ( X ) . In particular, the strong topology β ( E ( X ) , E ( X ) n ∼ ) ( E ( X ) n ∼ = the order continuous dual of E ( X ) ) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.

Structural aspects of truncated archimedean vector lattices: good sequences, simple elements

Richard N. Ball (2021)

Commentationes Mathematicae Universitatis Carolinae

The truncation operation facilitates the articulation and analysis of several aspects of the structure of archimedean vector lattices; we investigate two such aspects in this article. We refer to archimedean vector lattices equipped with a truncation as truncs. In the first part of the article we review the basic definitions, state the (pointed) Yosida representation theorem for truncs, and then prove a representation theorem which subsumes and extends the (pointfree) Madden representation theorem....

Structure of Cesàro function spaces: a survey

Sergey V. Astashkin, Lech Maligranda (2014)

Banach Center Publications

Geometric structure of Cesàro function spaces C e s p ( I ) , where I = [0,1] and [0,∞), is investigated. Among other matters we present a description of their dual spaces, characterize the sets of all q ∈ [1,∞] such that C e s p [ 0 , 1 ] contains isomorphic and complemented copies of l q -spaces, show that Cesàro function spaces fail the fixed point property, give a description of subspaces generated by Rademacher functions in spaces C e s p [ 0 , 1 ] .

Structure of Rademacher subspaces in Cesàro type spaces

Sergey V. Astashkin, Lech Maligranda (2015)

Studia Mathematica

The structure of the closed linear span of the Rademacher functions in the Cesàro space C e s ∞ is investigated. It is shown that every infinite-dimensional subspace of either is isomorphic to l₂ and uncomplemented in C e s ∞ , or contains a subspace isomorphic to c₀ and complemented in . The situation is rather different in the p-convexification of C e s ∞ if 1 < p < ∞.

Structure of spaces of germs of holomorphic functions.

Nguyen Van Khue, P. Thien Danh (1997)

Publicacions Matemàtiques

Let E be a Frechet (resp. Frechet-Hilbert) space. It is shown that E ∈ (Ω) (resp. E ∈ (DN)) if and only if [H(OE)]' ∈ (Ω) (resp. [H(OE)]' ∈ (DN)). Moreover it is also shown that E ∈ (DN) if and only if Hb(E') ∈ (DN). In the nuclear case these results were proved by Meise and Vogt [2].

Structure spaces for rings of continuous functions with applications to realcompactifications

Lothar Redlin, Saleem Watson (1997)

Fundamenta Mathematicae

Let X be a completely regular space and let A(X) be a ring of continuous real-valued functions on X which is closed under local bounded inversion. We show that the structure space of A(X) is homeomorphic to a quotient of the Stone-Čech compactification of X. We use this result to show that any realcompactification of X is homeomorphic to a subspace of the structure space of some ring of continuous functions A(X).

Subelliptic Poincaré inequalities: the case p &lt; 1.

Stephen M. Buckley, Pekka Koskela, Guozhen Lu (1995)

Publicacions Matemàtiques

We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p &lt; 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.

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