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 $\lambda \left(v\right)=su{p}_X\lambda (v;X)$. The purpose of this paper is to recapture this result by exhibiting a simple formula for a subspace V contained in $L^{\infty }\left(\nu \right)$ and isometric to v and a projection $P_{\infty }$ from C ⊕ V onto V such that $\parallel P_{\infty }\parallel =\parallel P₁\parallel$, where P₁ is a minimal projection from L¹(ν) onto v. Specifically, if $P₁={\sum }_{i=1}^2{U}_i\otimes v_i$, then $P_{\infty }={\sum }_{i=1}^2{u}_i\otimes V_i$, where $d{V}_i=2v_{i}d\nu$ and $d{U}_i=-2u_{i}d\nu$.

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.

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...

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.

Let X be a normed space and $G_F\left(X\right)$ the group of all linear surjective isometries of X that are finite-dimensional perturbations of the identity. We prove that if $G_F\left(X\right)$ acts transitively on the unit sphere then X must be an inner product space.

We define a handy new modulus for normed spaces. More precisely, given any normed space X, we define in a canonical way a function ξ:[0,1)→ ℝ which depends only on the two-dimensional subspaces of X. We show that this function is strictly increasing and convex, and that its behaviour is intimately connected with the geometry of X. In particular, ξ tells us whether or not X is uniformly smooth, uniformly convex, uniformly non-square or an inner product space.

This note is an announcement of results contained in the papers [4], [5], [6] concerning isomorphic properties of Banach spaces in projective tensor products (for this definition and some property we refer to [1]). At the end, some new result is obtained too.

Si prendono in considerazione particolari costanti relative alla struttura della sfera unitaria di uno spazio di Banach. Se ne studiano alcune generali proprietà, con particolare riferimento alle relazioni con il modulo di convessità dello spazio. Se ne fornisce inoltre una esatta valutazione negli spazi $l_p$.

A sequence (xn) in a Banach space X is said to be weakly-p-summable, 1 ≤ p &lt; ∞, when for each x* ∈ X*, (x*xn) ∈ lp. We shall say that a sequence (xn) is weakly-p-convergent if for some x ∈ X, (xn - x) is weakly-p-summable.

In this note we review some results about:1. Representation of Absolutely (∞,p) summing operators (∏∞,p) in C(K,E)2. Dunford-Pettis properties.

We investigate almost ball remotal and ball remotal subspaces of Banach spaces. Several subspaces of the classical Banach spaces are identified having these properties. Some stability results for these properties are also proved.

We investigate Hyers-Ulam stability of non-surjective ε-isomeries of Banach spaces. We also pose and discuss an open problem.

Every separable infinite-dimensional superreflexive Banach space admits an equivalent norm which is Fréchet differentiable only on an Aronszajn null set.

An alternative to the Dunford-Pettis Property, called the DP1-property, is introduced. Its relationship to the Dunford-Pettis Property and other related properties is examined. It is shown that ${\ell }_p$-direct sums of spaces with DP1 have DP1 if 1 ≤ p < ∞. It is also shown that for preduals of von Neumann algebras, DP1 is strictly weaker than the Dunford-Pettis Property, while for von Neumann algebras, the two properties are equivalent.

We introduce a weaker version of the polynomial Daugavet property: a Banach space X has the alternative polynomial Daugavet property (APDP) if every weakly compact polynomial P: X → X satisfies $ma{x}_{\omega \in }\left|\right|Id+\omega P\left|\right|=1+\left|\right|P\left|\right|$. We study the stability of the APDP by c₀-, ${\ell }_{\infty }$- and ℓ₁-sums of Banach spaces. As a consequence, we obtain examples of Banach spaces with the APDP, namely $L_{\infty }(\mu ,X)$ and C(K,X), where X has the APDP.