‎Let $X$ be a real normed  space, then  $C(subseteq X)$  is  functionally  convex  (briefly, $F$-convex), if  $T(C)subseteq Bbb R $ is  convex for all bounded linear transformations $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. We improve the    Krein-Milman theorem  ...

Let Ω ⊂ R be a bounded, convex domain. We study the best constant of the Sobolev trace embedding W 1,∞(Ω) ↪→ L∞(∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole. That is, we deal with the minimization problem S A = inf ‖u‖W1,∞(Ω)/‖u‖L∞(∂Ω) for functions that verify u |A= 0. We find that there exists an optimal hole that minimizes the best constant S A among subsets of Ω o...

Let X be a Banach space and Z a nonempty closed subset of X. Let J :Z → R be an upper semicontinuous function bounded from above. This paper is concerned with the perturbed optimization problem supz∈Z{J (z)+ ‖x − z‖}, which is denoted by (x, J )-sup. We shall prove in the present paper that if Z is a closed boundedly relatively weakly compact nonempty subset, then the set of all x ∈X for which ...

In this paper, we will provide some examples of Banach spaces that are Gâteaux differentiability spaces but not weak Asplund, weak Asplund but not in class(
̃), in class(
̃) but whose dual space is not weak∗ fragmentable. We begin with some definitions. A Banach space X is called a weak Asplund space [almost weak Asplund] (Gâteaux differentiability space) if each continuous convex function define...

We present the material in a slightly different order than it is usually done (such as e.g. in the course book). Here we prefer to start out with an abelian C∗-algebra A (say, the algebra C∗(a) generated by a normal operator a ∈ B(H)) and construct from it the spectral measure. This is done as follows: We know that A is isomorphic to C(X), the algebra of continuous functions on a compact set X....

A subset of vertices of a graph is called a dominating set if every vertex of the graph which is not present in the set has at least one neighbour in it. Dominating set polytope of a graph is defined as the convex hull of 0/1-incidence vectors of all the dominating sets of the graph. This paper presents complete characterization of the dominating set polytope of a cycle.

If K is a pre-compact subset of a normed space X and > 0, the quantity logN(K, BX) describes the complexity of K at the level of resolution . A 1972 conjecture of A. Pietsch – originally stated in the operatortheoretic laguage – asserts that the two “metric entropy functionals,” → logN(K, BX) and → logN(BX∗ , K◦), are equivalent in the appropriate sense, uniformly over normed spaces X and over ...

The object of this note is to give two applications of an intersection lemma of Ky Fan. First it is used to obtain a variational property of a strongly continuous function on a weakly compact convex subset of a normed space. In the second half we apply the lemma to obtain a direct proof of a result on the extension of monotone sets in topological linear spaces. It was established separately by ...

Let E be a real Banach space, let C be a closed convex subset of E, let T be a nonexpansive mapping of C into itself, that is, ‖Tx−Ty‖ ≤ ‖x− y‖ for each x, y ∈ C, and let A⊂ E×E be an accretive operator. For r > 0, we denote by Jr the resolvent of A, that is, Jr = (I + rA)−1. The problem of finding a solution u∈ E such that 0∈ Au has been investigated by many authors; for example, see [3, 4, 7,...

It is not difficult to see that every group homomorphism from Z to R extends to a homomorphism from R to R. We discuss other examples of discrete subgroups Γ of connected Lie groups G, such that the homomorphisms defined on Γ can (“virtually”) be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that Γ has this property if it is Zariski dens...