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S. Troyanski

Publications and source records attributed to S. Troyanski.

6 recordsLinked to original sources
arXiv2018

Locally uniformly rotund renormings of the spaces of continuous functions on Fedorchuk compacts

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such that $f^{-1}(y)$ is metrizable for all $y\in Y$ . A continuous map of compacts $f : X \to Y$ is said to be fully closed if for any disjoint closed subsets $A;B \subset X$ the intersection $f(A) \cap f(B)$ is finite. For instance the projection of the lexicographic square onto the first factor is fully closed and all its fibers are homeomorphic to the closed interval.

math.FA↗
arXiv2013

Polyhedrality in Pieces

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 hypotheses of both tools are based on countable decompositions, either of the unit sphere S_X or of certain subsets of the dual ball of a given Banach space X. The sufficient conditions of Theorem 4 are shown to be necessary in the separable case. Using Theorem 7, we can unify two known results regarding the polyhedral renorming of certain C(K) spaces, and spaces having an (uncountable) unconditional basis. New examples of spaces having equivalent polyhedral norms are given in the final section.

math.FA↗
arXiv2011

Moduli of convexity and smoothness of reflexive subspaces of L^1

We show that for any probability measure μthere exists an equivalent norm on the space L^1(μ) whose restriction to each reflexive subspace is uniformly smooth and uniformly convex, with modulus of convexity of power type 2. This renorming provides also an estimate for the corresponding modulus of smoothness of such subspaces.

math.FA↗
arXiv2008

A note on fragmentability and weak-G_delta sets

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 separable polyhedral space.

math.FA↗