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

Finite Ball Intersection Property of the Urysohn Universal Space

In a paper published posthumously, P.S. Urysohn constructed a complete, separable metric space that contains an isometric copy of every complete separable metric space, nowadays referred to as the Urysohn universal space. Here we study various convexity properties of the Urysohn universal space and show that it has a finite ball intersection property. We also note that Urysohn universal space is not hyperconvex

Approximation Schemes, Related S-Numbers and Applications.

A generalized approximation scheme via a sequence (p(,n)) of properties on a Banach space X is introduced. Approximation numbers and Kolmogorov diameters are defined in this context and comparison of approximation numbers and Kolmogorov diameters of T (ELEM) L(X) with those of T' and JT are studied. Compact sets and compact maps are defined with respect to this approximation scheme (we call them Q-compact sets and Q-compact maps) and a Dieudonne-Schwartz-type characterization of Q-compact sets is obtained. A relation between Q-compact sets and Q-compact maps and a representation theorem for Q-compact maps are proved. Q-compact maps are genuine generalization of compact maps is shown by means of an example of a Q-compact map which is not a compact map. Replacing the role of l(,u) in the classical approximation spaces with a nuclear infinite type power series space an approximation space is defined. Representation and transformation theorems for such spaces are obtained. Also, using K and J-functionals and stable nuclear infinite type power series spaces, discrete intermediate spaces are defined and their interpolation are examined. The notion of an operator measure s on L(X) is defined and a formula for the essential spectral radius r(,e)(T) is obtained in terms of (s(T('n))).Ph.D.MathematicsUniversity of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/160252/1/8502751.pd