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In this paper, we present two new fixed point theorems in Fr?chet algebras and Fr?chet spaces. Our fixed point results are expressed with the help of family of measures of noncompactness and generalizes Darbo theorem. As an application, we establish some existence results for various types of nonlinear integral equations.

In this paper we define bornological and b-bornological convergence in locally convex cones. We give some characterizations for these convergences. In special case of locally convex topological vector spaces, both these concepts reduce to the known concept of Mackey-convergence. We investigate the relationship between bornological cones and bornological convergence. Finally, we investigate bornological completeness.

The weak and weak* topologies are introduced for Banach spaces and their duals, and the Banach-Alaoglu theorem is proved. Further topologies are introduced on the spaces of bounded linear operators, and these are placed in the context of the general locally convex vector spaces. Properties of convex bodies are discussed, including the Krein-Milman and Choquet theorems, and several applications of weak* compactness are given. These include Furstenberg’s proof of Weyl’s polynomial equidistribution theorem and elliptic regularity for the Laplace operator at the boundary. The theory of distributions is introduced.

A Lebesgue-type integration theory in complete bornological locally convex topological vector spaces was introduced by the first author in [17]. In this paper we continue developing this integration technique and formulate and prove some theorems on integrable functions as well as some convergence theorems. An example of Dobrakov integral in non-metrizable complete bornological locally convex spaces is given.

We propose a definition for strict inductive limits in locally convex cones. By this definition, we prove that the strict inductive limit of a sequence of locally convex cones with the strict separation property has the same strict separation property. Also we establish that the strict inductive limit of a sequences of separated cones is separated too. Finally we verify barreledness for this strict inductive limit.

The bornological product measures via the generalized Dobrakov integral in complete bornological locally convex spaces are studied using the domination of considered vector measures. A Fubini-type theorem for such product measures is proven.

Weak and strong continuity conditions for representations of topological groups in locally convex spaces are considered. In particular, weak continuity conditions of reducible locally equicontinuous representations of a topological group in a locally convex space that define weakly continuous representations in an invariant subspace and in the quotient space by this invariant subspace are investigated. These conditions help one to prove the weak continuity of averages and approximations related to weakly continuous locally equicontinuous quasirepresentations of amenable topological groups. Strong continuity conditions for a representation approximating a quasirepresentation of this kind are related to conditions of automatic strong continuity of weakly continuous representations, and fail to hold for some groups, spaces, and representations. In this connection, strong continuity conditions for weakly continuous representations in quasicomplete barrelled locally convex spaces are indicated for a broad class of topologically pseudocomplete groups (which includes the Cech complete groups and the locally pseudocompact groups). Several examples are discussed, in particular, ones relating to the construction of -products with distinguished subgroups.

For all linear and many nonlinear operators between Frechet spaces, several continuity results are established. Especially, a new closed graph theorem is given.

In this article, we present a method to obtain new Ekeland-type variational principles for mappings defined between locally convex spaces. The method is based on a special nuclear cone defined in a product of two locally convex spaces and on Pareto efficiency with respect to this cone.

An inductive locally convex limit of reflexive topological spaces is reflexive iff it is almost regular.

Spaces of entire functions of several complex variables occupy an important position in view of their vast applications in various branches of mathematics, for instance, the classical analysis, theory of approximation, theory of topological bases etc. With an idea of correlating entire functions with certain aspects in the theory of basis in locally convex spaces, we have investigated in this paper the bornological aspects of the space $X$ of integral functions of several complex variables. By $Y$ we denote the space of all power series with positive radius of convergence at the origin. We introduce bornologies on $X$ and $Y$ and prove that $Y$ is a convex bornological vector space which is the completion of the convex bornological vector space $X$.

We shall construct in ZFC two Frechet–Urysohn α 4 -spaces, the product of which is α 4 , but fails to be Frechet–Urysohn. This answers Nogura's question from 1985.

A generalization of I. Dobrakov's integral to complete bornological locally convex spaces is given.

Countable spaces with an only nonisolated point for which the free topological groups are the inductive limits of the sets of words having restricted length are characterized. It is proved for a collectionwise normal X that if its free topological group is the inductive limit of the sets of words having restricted length, then any closed discrete set of non-P-points in X is countable. All the results obtained are valid for both Abelian and non-Abelian free groups.

A strictly barrelled diskBin a Hausdorff locally convex spaceEis a disk such that the linear span ofBwith the topology of the Minkowski functional ofBis a strictly barrelled space. Valdivia's closed graph theorems are used to show that closed strictly barrelled disk in a quasi-(LB)-space is bounded. It is shown that a locally strictly barrelled quasi-(LB)-space is locally complete. Also, we show that a regular inductive limit of quasi-(LB)-spaces is locally complete if and only if each closed bounded disk is a strictly barrelled disk in one of the constituents.

A notion of an almost regular inductive limits is introduced. Every sequentially complete inductive limit of arbitrary locally convex spaces is almost regular.

In this paper we give a characterization of p-adic semi-Montel spaces which allows us to describe the finest polar semi-Montel topology coarser than the original topology of a locally convex space. As an application we derive that every polar semi-Montel space is a polar inductive limit of a family of nuclear spaces. We also pay attention to the connection with compactifying operators.

We study Dieudonne-Kothe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.

Lebesgue’s differentiation theorem (LDT) states that every monotonic real function is differentiable a.e. We investigate the validity of this theorem for functions with values in topological vector lattices. It is shown that a Fréchet lattice satisfies (LDT) iff it is isomorphic to a generalized echelon space, a Banach lattice satisfies (LDT) iff it is isomorphic to some l 1 ( Γ ) {l^1}\left ( \Gamma \right ) .

The object of this paper is to study the stability of the quasi-completeness of theLα spaces for locally convex vector valued functions, by the inductive limits and the countable products.

A Fréchet space is called twisted if it is not isomorphic to a countable product of Fréchet spaces with continuous norms. It is easy to show that no non-archimedean Fréchet space with a Schauder basis is twisted. We construct examples of non-archimedean twisted nuclear Fréchet 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.

Let $Q$ be a bounded, convex and locally closed subset of \ $\C^N$, let $H(Q)$ be the space of all functions which are holomorphic on an open neighborhood of $Q$. We endow $H(Q)$ with its projective topology. We show that the topology of the weighted inductive limit of Fr\'echet spaces of entire functions which is obtained as the Laplace transform of the strong dual to $H(Q)$ can be described be means of canonical weighted seminorms if and only if the intersection of $Q$ with each supporting hyperplane to the closure of $Q$ is compact. We also find conditions under which this (LF)-space of entire functions coincides algebraically with its projective hull.

There are obtained the sufficient conditions of a termwise differentiation for the series of differentiable mappings from a segment into the linear inductive scale of the locally convex spaces and the analogous ones for the series of homogeneously differentiable mappings from a segment into the arbitrary inductive scale of the locally convex spaces. Some examples are considered.

Local convexifiability, LC, is studied for generic smooth vector functions and vector functions with small codimension singularities. We discuss its connection with Pareto optima and describe the global structure of manifolds admitting generic locally convex vector functions. We describe also the spaces of LC vector functions.

Certain locally convex spaces of scalar-valued mappings are shown to be finite-dimensional.