scholarly journals Tilings of Normed Spaces

2017 ◽  
Vol 69 (02) ◽  
pp. 321-337 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Libor Veselý
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
The Other ◽  
Locally Convex Spaces ◽  
List Type ◽  
Topological Linear Space ◽  
Normed Spaces ◽  
Infinite Dimensional ◽  
Topological Linear ◽  
Locally Convex

Abstract By a tiling of a topological linear space X, we mean a covering of X by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite dimensional spaceswas initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study the existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space X, our main results are the following. (i) X admits no tiling by Fréchet smooth bounded tiles. (ii) If X is locally uniformly rotund (LUR), it does not admit any tiling by balls. (iii) On the other hand, some spaces, г uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

The graph-theoretic analogue of Tietze's characterization of a convex set and its generalization

1972 ◽  
Vol 72 (1) ◽  
pp. 7-9
Author(s):  
M. D. Guay
Keyword(s):  
Linear Space ◽  
Closed Subset ◽  
Convex Sets ◽  
Convex Set ◽  
Topological Linear Space ◽  
Connected Subset ◽  
Graph Theoretic ◽  
Topological Linear ◽  
Locally Convex

Introduction. One of the most satisfying theorems in the theory of convex sets states that a closed connected subset of a topological linear space which is locally convex is convex. This was first established in En by Tietze and was later extended by other authors (see (3)) to a topological linear space. A generalization of Tietze's theorem which appears in (2) shows if S is a closed subset of a topological linear space such that the set Q of points of local non-convexity of S is of cardinality n < ∞ and S ~ Q is connected, then S is the union of n + 1 or fewer convex sets. (The case n = 0 is Tietze's theorem.)

scholarly journals On reducibility of some operator semigroups and algebras on locally convex spaces

2005 ◽  
Vol 2005 (24) ◽  
pp. 3951-3961
Author(s):  
Edvard Kramar
Keyword(s):  
Locally Convex Spaces ◽  
Normed Spaces ◽  
Operator Semigroups ◽  
Locally Convex ◽  
Convex Spaces

A generalization of some results from normed spaces, concerning reducibility and triangularizability of semigroups and algebras of operators, to locally convex spaces is given.

Separating Translates in the Plane: Combinatorial Bounds and an Algorithm

1997 ◽  
Vol 07 (06) ◽  
pp. 551-562
Author(s):  
Jurek Czyzowicz ◽  
Hazel Everett ◽  
Jean-Marc Robert
Keyword(s):  
Pairwise Disjoint ◽  
Convex Sets ◽  
Convex Set ◽  
Time Algorithm ◽  
The Other ◽  
Separation Problem ◽  
Disjoint Convex

Given a family of pairwise disjoint convex sets S in the plane, a set [Formula: see text] is separated from a subset X ⊆ S if there exists a line l such that [Formula: see text] lies on one side of l and the sets in X lie on the other side. In this paper, we establish two combinatorial bounds related to the separation problem for families of n pairwise disjoint translates of a convex set in the plane: 1) there exists a line which separates one translate from at least [Formula: see text] translates, for some constant [Formula: see text] that depends only on the shape of the translates and 2) there is a function [Formula: see text], defined only by the shape of [Formula: see text], such that there exists a line with orientation Θ or [Formula: see text] which separates one translate from at least [Formula: see text] translates, for any orientation Θ. We also present an O(n log (n + k) + k) time algorithm for finding a translate which can be separated from the maximum number of translates amongst families of n pairwise disjoint translates of convex k-gons in the plane.

COMPATIBLE LOCALLY CONVEX TOPOLOGIES ON NORMED SPACES: CARDINALITY ASPECTS

2017 ◽  
Vol 96 (1) ◽  
pp. 139-145 ◽  
Author(s):  
ELENA MARTÍN-PEINADOR ◽  
ANATOLIJ PLICHKO ◽  
VAJA TARIELADZE
Keyword(s):  
Dimensional Space ◽  
Infinite Dimensional Space ◽  
Normed Spaces ◽  
Norm Topology ◽  
Infinite Dimensional ◽  
The Family ◽  
Locally Convex Topologies ◽  
Original Norm ◽  
Locally Convex

For a normed infinite-dimensional space, we prove that the family of all locally convex topologies which are compatible with the original norm topology has cardinality greater or equal to $\mathfrak{c}$.

A Tauberian theorem for statistical convergence

1989 ◽  
Vol 106 (2) ◽  
pp. 277-280 ◽  
Author(s):  
I. J. Maddox
Keyword(s):  
Linear Space ◽  
Tauberian Theorem ◽  
Statistical Convergence ◽  
Slow Oscillation ◽  
Topological Linear Space ◽  
Tauberian Condition ◽  
Topological Linear ◽  
Locally Convex

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

Infinite Doubly Stochastic Matrices

1962 ◽  
Vol 5 (1) ◽  
pp. 1-4 ◽  
Author(s):  
J.R. Isbell
Keyword(s):  
Linear Space ◽  
Convex Set ◽  
The Other ◽  
Locally Convex Spaces ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Incorrect Assertion ◽  
Doubly Stochastic Matrices ◽  
The Axiom Of Choice ◽  
Locally Convex

This note proves two propositions on infinite doubly stochastic matrices, both of which already appear in the literature: one with an unnecessarily sophisticated proof (Kendall [2]) and the other with the incorrect assertion that the proof is trivial (Isbell [l]). Both are purely algebraic; so we are, if you like, in the linear space of all real doubly infinite matrices A = (aij).Proposition 1. Every extreme point of the convex set of ail doubly stochastic matrices is a permutation matrix.Kendall's proof of this depends on an ingenious choice of a topology and the Krein-Milman theorem for general locally convex spaces [2]. The following proof depends on practically nothing: for example, not on the axiom of choice.

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