Bernstein's Lethargy Theorem in Frechet Spaces

In this paper we consider Bernstein's Lethargy Theorem (BLT) in the context of Fr\'{e}chet spaces. Let $X$ be an infinite-dimensional Fr\'echet space and let $\mathcal{V}=\{V_n\}$ be a nested sequence of subspaces of $X$ such that $\bar{V_n} \subseteq V_{n+1}$ for any $n \in \mathbb{N}$ and $X=\bar{\bigcup_{n=1}^{\infty}V_n}.$ Let $e_n$ be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on \sup\{\{dist}(x, V_n)\}, we prove that there exists $x \in X$ and $n_o \in \mathbb{N}$ such that \frac{e_n}{3} \leq \{dist}(x,V_n) \leq 3 e_n for any $n \geq n_o$. By using the above theorem, we prove both Shapiro's \cite{Sha} and Tyuremskikh's \cite{Tyu} theorems for Fr\'{e}chet spaces. Considering rapidly decreasing sequences, other versions of the BLT theorem in Fr\'{e}chet spaces will be discussed. We also give a theorem improving Konyagin's \cite{Kon} result for Banach spaces.Comment: 20 page

Characterization Conditions and the Numerical Index

In this paper we survey some recent results concerning the numerical index $n(\cdot)$ for large classes of Banach spaces, including vector valued $\ell_p$-spaces and $\ell_p$-sums of Banach spaces where $1\leq p < \infty$. In particular by defining two conditions on a norm of a Banach space $X$, namely a Local Characterization Condition (LCC) and a Global Characterization Condition (GCC), we are able to show that if a norm on $X$ satisfies the (LCC), then $n(X) = \displaystyle\lim_m n(X_m).$ For the case in which $\mathbb{N}$ is replaced by a directed, infinite set $S$, we will prove an analogous result for $X$ satisfying the (GCC). Our approach is motivated by the fact that $n(L_p(\mu, X))= n(\ell_p(X)) = \displaystyle \lim_m n(\ell_p^m (X))$ \cite {aga-ed-kham}.Comment: 17 pages. arXiv admin note: text overlap with arXiv:1106.482

Asymptotic estimate of absolute projection constants

In this note we construct a sequence of real, k-dimensional symmetric spaces Y k satisfying lim inf k Sk = p k lim inf k (Y k; l1)= p k > max w2[0;a2] h(w) > 1=(2 p 2= ); where Sk is de ned by (4) and h(w) = a21 p 2= + 2a1 q a22 w2 + w q a22 w2 with a1 = 1=(2 p 2= ) and a2 = 1 a1. This improves the lower bound obtained in [3], Th. 5.3 by maxw2[0;a2] h(w)

On the dimension of the set of minimal projections

Let $X$ be a finite-dimensional normed space and let $Y \subseteq X$ be its proper linear subspace. The set of all minimal projections from $X$ to $Y$ is a convex subset of the space all linear operators from $X$ to $X$ and we can consider its affine dimension. We establish several results on the possible values of this dimension. We prove optimal upper bounds in terms of the dimensions of $X$ and $Y$. Moreover, we improve these estimates in the polyhedral normed spaces for an open and dense subset of subspaces of the given dimension. As a consequence, in the polyhedral normed spaces a minimal projection is unique for an open and dense subset of hyperplanes. To prove this, we establish certain new properties of the Chalmers-Metcalf operator. Another consequence is the fact, that for every subspace of a polyhedral normed space, there exists a minimal projection with many norming pairs. Finally, we provide some more refined results in the hyperplane case.Comment: 31 page

Sequence Lorentz spaces and their geometric structure

This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity, and strict convexity in the sequence Lorentz spaces γp,w. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space γ1,w. We also establish a complete description up to isometry of the dual and predual spaces of the sequence Lorentz spaces γ1,w written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental application of geometric structure of γ1,w to one-complemented subspaces of γ1,w