Arbitrary Topological Vector Space Research Articles

Gateaux Differentiability Revisited

We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided.

Intersection Theorems for Closed Convex Sets and Applications

A number of landmark existence theorems of nonlinear functional analysis follow in a simple and direct way from the basic separation of convex closed sets in finite dimension via elementary versions of the Knaster-Kuratowski-Mazurkiewicz principle - which we extend to arbitrary topological vector spaces - and a coincidence property for so-called von Neumann relations. The method avoids the use of deeper results of topological essence such as the Brouwer Fixed Point Theorem or the Sperner's Lemma and underlines the crucial role played by convexity. It turns out that the convex KKM Principle is equivalent to the Hahn-Banach Theorem, the Markov-Kakutani Fixed Point Theorem, and the Sion-von Neumann Minimax Principle.

Integration of ε-Fenchel Subdifferentials and Maximal Cyclic Monotonicity

This paper concerns the integration of ?-Fenchel subdifferentials of proper lower semicontinuous convex functions defined on arbitrary topological vector spaces. We make use of integration tools to provide a representation formula of the approximate subdifferential of convex functions, and also to identify the class of maximal cyclically monotone families of operators.

On C-sequences of operators

Invariant results are established for a considerably general multiplier convergence of operator series where the operators are defined on arbitrary topological vector spaces and valued in arbitrary locally convex spaces.

Some fixed point theorems for nonconvex spaces

We give a theorem for nonconvex topological vector spaces which yields the classical fixed point theorems of Ky Fan, Kim, Kaczynski, Kelly and Namioka as immediate consequences, and prove a new fixed point theorem for set‐valued maps on arbitrary topological vector spaces.

Remarks on regular (LF)-spaces

Developing ideas found in a recent paper of Gilsdorf to arbitrary topological vector spaces (tvs) one shows that a Hausdorff (LB)tv-spaceE is regular provided every null-sequence inE has a series convergent subsequence inE.

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L ∞( M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's ( J. Econ. Theory 4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.

Linear operators whose domain is locally convex

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 S 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 is quasi-convex, where U is the unit ball of X.

Exhaustive measures in arbitrary topological vector spaces

Exhaustive measures in arbitrary topological vector spaces

The nonstandard theory of topological vector spaces

The nonstandard theory of topological vector spaces

Analytic functions and the Fourier transform of distributions. II

1. Introduction. The present paper is a continuation of the work of Part I (see [5 ]) in which we showed how to define the Fourier transform of an arbitrary distribution of L. Schwartz. We were led, in a natural manner, to the consideration of linear functions on spaces of analytic functions. In ?2, below, we shall show how our previous methods apply also when dealing with indefinitely differentiable maps on Euclidean space of compact carrier with values in an arbitrary topological vector space V. The Fourier transform of such a map is an map of exponential of the Euclidean space into V, and any such entire map of exponential type is the Fourier transform of an entire map of compact carrier of the Euclidean space into V. Once this generalized Paley-Wiener theorem is established, all our previous method on the Fourier transform of distributions can be extended to this generalized situation. In this connection we should mention that there are two methods of defining vector-valued distributions; the general kernel theorem of L. Schwartz states that these two methods yield the same result. We present in ?2 a simple proof of this theorem. We consider also in ?2 the concepts of an analytic function on a topological vector space, and, more generally, of an analytic map of one topological vector space into another. Our notion of analytic map is a generalization of analyticity for Banach spaces (see [6, Chapter 41). We expect to use analytic functions on the space D (see [5; 7 ]) to extend the theory of distributions of L. Schwartz so as to apply to certain kinds of nonlinear differential equations. In certain cases we can prove that our analytic functions are limits of polynomials (i.e. belong to the ring generated by constants and linear functions). This situation is markedly different from that for infinite dimensional Banach spaces. In ?3 below, we consider some applications of the Fourier transform of distributions to the study of functions analytic in the interior of a strip in the complex plane, and having a very general type of boundary value on the sides of the strip. We show that the functions considered are uniquely determined by their boundary values. We also establish a generalization of Cauchy's formula, by means of which we establish a generalization of the following theorem of Phragmen and Lindelof: Let F(z) be analytic in the