Conjecture [Schauder] For every non-empty convex subset C of a topological vector space E, a compact continuous mapping f : C → C has a fixed point, i.e., a point x∗ ∈ C such that f(x∗) = x∗. (See [16], problem 54). We recall that a mapping f : C → C is said to be compact if f(C) is contained in a compact subset of C. Schauder proved in 1930 that his conjecture holds for normed vector spaces an...

,ABSTRACT. There is a formula (Gelfand’s formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operator T defined on a complete topological vector space, locally convex. We also show an easy way to find a n...

Let F be an arbitrary topological vector space; we shall say that a subset S of F is quasi-convex if the set of continuous affine functionals on 5 separates the points of S. If X is a Banach space and T: X -* F is a continuous linear operator, then T is quasi-convex if T(U) is quasiconvex, where U is the unit ball of X. In the case when T is compact, T(U) is quasi-convex if and only if it is af...

Let G be a maximally almost periodic (MAP) Abelian group and let B be a boundedness on G in the sense of Vilenkin. We study the relations between B and the Bohr topology of G for some well known groups with boundedness (G,B). As an application, we prove that the Bohr topology of a topological group which is topologically isomorphic to the direct product of a locally convex space and an L∞-group...

in this paper, the concept of {sl local base with  stratifiedstructure} in $i$-topological vector spaces is introduced. weprove that every $i$-topological vector space has a balanced localbase with stratified structure. furthermore, a newcharacterization of $i$-topological vector spaces by means of thelocal base with stratified structure is given.

In this paper, we investigate the concept of topological stationary for locally compact semigroups. In [4], T. Mitchell proved that a semigroup S is right stationary if and only if m(S) has a left Invariant mean. In this case, the set of values ?(f) where ? runs over all left invariant means on m(S) coincides with the set of constants in the weak* closed convex hull of right translates of f. Th...