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In this paper we establish the principle of uniform boundedness for LSC convex processes in some class of locally convex spaces (strictly N locally convex spaces). Thus, we generalize the same result established by S. Lahrech in [1] for sequentially continuous linear operators.
Content available remote Sylvester inertia law in commutative Leibniz algebras with logarithms
In algebras with logarithms induced by a given right invertible operator D one can define quadratic forms by means of power mappings induced by logarithmic mapping. Main results of this paper will be concerned with the case when an algebra X under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy+yDx for x, y is an element of dom D. If X is an locally m-convex algebra then these forms have the similar properties as quadratic forms in the Euclidian spaces En, including the Sylvester inertia law.
In this paper we introduce some new sequence spaces by using a sequence of moduli F = (fk), give some topological properties and inclusion relations related to these sequence spaces. We also give the beta-dual of [�c, F, p]infinity(delta m).
Content available remote A Kneser-type theorem for an integral equation in locally convex spaces
We shall give suffcient conditions for the existence of solutions of the integral equation (1) in locally convex spaces. We also prove that the set of these solutions is a continuum.
According to Mickael's selection theorem any surjective continuous linear operator from one Prechet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if E is a Frechet space and T : E -> E is a continuous linear operator such that the Cauchy problem x = T x, x(0) = X0 is solvable in [0,1] for any X06 E, then for anyf zawiera się C([0, 1],E), there exists a continues map S : [0,1] x E -> E, (t x) ->o StX such that for any X0 zawiera się w E, the function x(t) = StX0 is a solution of the Cauchy problem x(t) = Tx(t) +- f(t), x(0) = X0 (they call S a fundamental system of solutions of the equation x = Tx + f). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Frechet spaces and strong duals of Frechet-Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.
Content available remote Fourier analysis on locally convex spaces of distributions, II
This is the second in a series of papers, extending the theory of Fourier analysis to locally convex spaces of distributions (LCD-spaces). In this paper, LCD-spaces admitting conjugation and multiplier operators on LCD-spaces are discussed. It is also shown that if E is an LCD-space having C°° as a dense subset, then E*, endowed with the topology of precompact convergence, is an LCD-space having C°° as a dense subset.
Content available remote Fourier analysis on locally convex spaces of distributions, I
In this series of papers, many results of Fourier analysis which are known for Lp (1 < p < oo) , C (the space of continuous functions) and other Banach spaces of functions have been generalized to locally convex spaces of distributions. Also, in this paper, the (C, 1)-complementary space E' of a locally convex space of distributions E is defined and it is shown that E' , as a subspace of E* with strong* topology, is a locally convex space of distributions.
Content available remote A characterization of complete Boolean algebras
Every lattice and, in particular, every Boolean algebra is a convexity space with a naturally defined convexity structure. We characterize complete Boolean algebras as the only S3 convexity spaces having an extension property for certain classes of convexity preserving maps. This answers our question posed in [1]. Our characterization provides also a short proof of Sikorski's extension theorem for homomorphisms of Boolean algebras.
Content available remote Extension theorems in axiomatic theory of convexity
We present a criterion for extending convexity preserving maps of convexity spaces. In a special case of convexity generated by a lattice structure this gives Sikorski's Extension Criterion for extending of maps of lattices. We also consider the class of convexity absolute extensors. It appears that complete Boolean algebras with a natural convexity belong to this class. In particular, we present an analogue of Tietze-Urysohn's Extension Theorem for maps of convexity spaces with values in a complete Boolean algebra.
Content available remote Strict 2-convexity and strict convexity
In this paper, we give some new characterizations of strict convexity and strict 2-convexity in linear 2-normed spaces.
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