22 citations · 82 across the 16 of their papers we have counts for
14 papers · 1 filter
A topological characterization of dual strict convexity in Asplund spaces
Richard J. Smith
Let be an Asplund space. We show that the existence of an equivalent norm on having a strictly convex dual norm is equivalent to the dual unit sphere (equivalentl…
Approximation of norms on Banach spaces
Richard J. Smith, Stanimir Troyanski
Relatively recently it was proved that if is an arbitrary set, then any equivalent norm on can be approximated uniformly on bounded sets by polyhedral norms and $C^\in…
Polyhedrality and decomposition
Trond A. Abrahamsen, Vladimir P. Fonf, Richard J. Smith +1
The aim of this note is to present two results that make the task of finding equivalent polyhedral norms on certain Banach spaces, having either a Schauder basis or an uncountable…
Topology, isomorphic smoothness and polyhedrality in Banach spaces
Richard J. Smith
In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called -locally relatively compact sets was i…
Smooth and polyhedral approximation in Banach spaces
Victor Bible, Richard J. Smith
We show that norms on certain Banach spaces can be approximated uniformly, and with arbitrary precision, on bounded subsets of by smooth norms and polyhedral n…
The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of
Eva Pernecká, Richard J. Smith
We prove that for certain subsets , , the Lipschitz-free space has the metric approximation property (MAP), with respect t…
Boundaries and polyhedral Banach spaces
Vladimir P Fonf, Richard J Smith, Stanimir Troyanski
We show that if and are Banach spaces, where is separable and polyhedral, and if is a bounded linear operator such that contains a boundary o…
Polyhedrality in Pieces
V. P. Fonf, A. J. Pallares, R. J. Smith +1
The aim of this paper is to present two tools, Theorems 4 and 7, that make the task of finding equivalent polyhedral norms on certain Banach spaces easier and more transparent. The…
Tree duplicates, -diagonals and Gruenhage spaces
Richard J. Smith
We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski an…
Operator machines on directed graphs
Petr Hajek, Richard J. Smith
We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| -->…
A note on fragmentability and weak-G_delta sets
V. P. Fonf, R. J. Smith, S. Troyanski
In terms of fragmentability, we describe a new class of Banach spaces which do not contain weak-G_delta open bounded subsets. In particular, none of these spaces is isomorphic to a…
Unconditional bases and strictly convex dual renormings
R. J. Smith, S. Troyanski
We present equivalent conditions for a space with an unconditional basis to admit an equivalent norm with a strictly convex dual norm.
Gruenhage compacta and strictly convex dual norms
Richard J. Smith
We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-cl…
Trees, linear orders and Gâteaux smooth norms
Richard J. Smith
We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on C(T), where T is a tree. This criterion is directly an…