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A simple formula showing L¹ is a maximal overspace for two-dimensional real spaces

B. L. Chalmers, F. T. Metcalf (1992)

Annales Polonici Mathematici

It follows easily from a result of Lindenstrauss that, for any real twodimensional subspace v of L¹, the relative projection constant λ(v;L¹) of v equals its (absolute) projection constant λ ( v ) = s u p X λ ( v ; X ) . The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in L ∞ ( ν ) and isometric to v and a projection P ∞ from C ⊕ V onto V such that ∥ P ∞ ∥ = ∥ P ₁ ∥ , where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if P ₁ = ∑ i = 1 2 U i ⊗ v i , then P ∞ = ∑ i = 1 2 u i ⊗ V i , where d V i = 2 v i d ν and d U i = - 2 u i d ν .

A simultaneous selection theorem

Alexander D. Arvanitakis (2012)

Fundamenta Mathematicae

We prove a theorem that generalizes in a way both Michael's Selection Theorem and Dugundji's Simultaneous Extension Theorem. We use it to prove that if K is an uncountable compact metric space and X a Banach space, then C(K,X) is isomorphic to C(𝓒,X) where 𝓒 denotes the Cantor set. For X = ℝ, this gives the well known Milyutin Theorem.

A space C(K) where all nontrivial complemented subspaces have big densities

Piotr Koszmider (2005)

Studia Mathematica

Using the method of forcing we prove that consistently there is a Banach space (of continuous functions on a totally disconnected compact Hausdorff space) of density κ bigger than the continuum where all operators are multiplications by a continuous function plus a weakly compact operator and which has no infinite-dimensional complemented subspaces of density continuum or smaller. In particular no separable infinite-dimensional subspace has a complemented superspace of density continuum or smaller,...

A stronger Dunford-Pettis property

H. Carrión, P. Galindo, M. L. Lourenço (2008)

Studia Mathematica

We discuss a strong version of the Dunford-Pettis property, earlier named (DP*) property, which is shared by both ℓ₁ and ℓ ∞ . It is equivalent to the Dunford-Pettis property plus the fact that every quotient map onto c₀ is completely continuous. Other weak sequential continuity results on polynomials and analytic mappings related to the (DP*) property are shown.

A strongly extreme point need not be a denting point in Orlicz spaces equipped with the Orlicz norm

Adam Bohonos, Ryszard Płuciennik (2011)

Banach Center Publications

There are necessary conditions for a point x from the unit sphere to be a denting point of the unit ball of Orlicz spaces equipped with the Orlicz norm generated by arbitrary Orlicz functions. In contrast to results in [12, 17, 16], we present also examples of Orlicz spaces in which strongly extreme points of the unit ball are not denting points.

A subsequence characterization of sequences spanning isomorphically polyhedral Banach spaces

G. Androulakis (1998)

Studia Mathematica

Let (x_n) be a sequence in a Banach space X which does not converge in norm, and let E be an isomorphically precisely norming set for X such that (*) ∑_n |x*(x_{n+1} - x_n)| < ∞, ∀x* ∈ E. Then there exists a subsequence of (x_n) which spans an isomorphically polyhedral Banach space. It follows immediately from results of V. Fonf that the converse is also true: If Y is a separable isomorphically polyhedral Banach space then there exists a normalized M-basis (x_n) which spans Y and there exists...

A survey on the Szlenk index and some of its applications.

Gilles Lancien (2006)

RACSAM

We describe how the Szlenk index has been used in various areas of the geometry of Banach spaces. We cover the following domains of application of this notion: non existence of universal spaces, linear classification of C(K) spaces, descriptive set theory, renorming problems and non linear classification of Banach spaces.

A survey on topological games and their applications in analysis.

Jiling Cao, Warren B. Moors (2006)

RACSAM

In this survey article we shall summarise some of the recent progress that has occurred in the study of topological games as well as their applications to abstract analysis. The topics given here do not necessarily represent the most important problems from the area of topological games, but rather, they represent a selection of problems that are of interest to the authors.

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