The Quantificational Tangent Cones

1988 ◽  
Vol 40 (3) ◽  
pp. 666-694 ◽  
Author(s):  
Doug Ward
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Nonsmooth Analysis ◽  
Tangent Cone ◽  
Building Blocks ◽  
Tangent Cones ◽  
Hausdorff Topological Vector Space ◽  
Local Approximations ◽  
Locally Convex

Nonsmooth analysis has provided important new mathematical tools for the study of problems in optimization and other areas of analysis [1, 2, 6-12, 28]. The basic building blocks of this subject are local approximations to sets called tangent cones.Definition 1.1. Let E be a real, locally convex, Hausdorff topological vector space (abbreviated l.c.s.). A tangent cone (on E) is a mapping A:2E × E → 2E such that A(C, x) is a (possibly empty) cone for all nonempty C in 2E and x in E.In the sequel, we will say that a tangent cone has a certain property (e.g. “A is closed” or “A is convex“) if A(C, x) has that property for all non-empty sets C and all x in C. (If A(C, x) is empty, it will be counted as having the property trivially.)

scholarly journals Asymptotic Farkas lemmas for convex systems

2016 ◽  
Vol 19 (4) ◽  
pp. 160-168
Author(s):  
Dinh Nguyen ◽  
Mo Hong Tran
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Constraint Qualification ◽  
Convex Set ◽  
Lower Semicontinuous ◽  
Closed Convex Cone ◽  
Convex Mapping ◽  
Hausdorff Topological Vector Space ◽  
Reverse Convex ◽  
Locally Convex

In this paper we establish characterizations of the containment of the set {xX: xC,g(x)K}{xX: f (x)0}, where C is a closed convex subset of a locally convex Hausdorff topological vector space, X, K is a closed convex cone in another locally convex Hausdorff topological vector space and g:X Y is a K- convex mapping, in a reverse convex set, define by the proper, lower semicontinuous, convex function. Here, no constraint qualification condition or qualification condition are assumed. The characterizations are often called asymptotic Farkas-type results. The second part of the paper was devoted to variant Asymptotic Farkas-type results where the mapping is a convex mapping with respect to an extended sublinear function. It is also shown that under some closedness conditions, these asymptotic Farkas lemmas go back to non-asymptotic Farkas lemmas or stable Farkas lemmas established recently in the literature. The results can be used to study the optimization

scholarly journals A NOTE ON MEASURE HOMOLOGY

2013 ◽  
Vol 56 (1) ◽  
pp. 87-92
Author(s):  
ROBERTO FRIGERIO
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Hyperbolic Manifolds ◽  
Simplicial Volume ◽  
Princeton University ◽  
And Topology ◽  
Hausdorff Topological Vector Space ◽  
Cw Complexes ◽  
Measure Homology ◽  
Locally Convex

AbstractMeasure homology was introduced by Thurston (W. P. Thurston, The geometry and topology of 3-manifolds, mimeographed notes (Princeton University Press, Princeton, NJ, 1979)) in order to compute the simplicial volume of hyperbolic manifolds. Berlanga (R. Berlanga, A topologised measure homology, Glasg. Math. J. 50 (2008), 359–369) endowed measure homology with the structure of a graded, locally convex (possibly non-Hausdorff) topological vector space. In this paper we completely characterize Berlanga's topology on measure homology of CW-complexes, showing in particular that it is Hausdorff. This answers a question posed by Berlanga.

scholarly journals Generalizations of F. E. Browder's sharpened form of the schauder fixed point theorem

1987 ◽  
Vol 42 (3) ◽  
pp. 390-398 ◽  
Author(s):  
Kok-Keong Tan
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Convex Subset ◽  
Compact Convex ◽  
Compact Convex Subset ◽  
Hausdorff Topological Vector Space ◽  
Coincidence Theorem ◽  
Nonempty Compact ◽  
Nonempty Compact Convex Subset ◽  
Locally Convex

AbstractLet E be a Hausdorff topological vector space, let K be a nonempty compact convex subset of E and let f, g: K → 2E be upper semicontinuous such that for each x ∈ K, f(x) and g(x) are nonempty compact convex. Let Ω ⊂ 2E be convex and contain all sets of the form x − f(x), y − x + g(x) − f(x), for x, y ∈ K. Suppose p: K × Ω →, R satisfies: (i) for each (x, A) ∈ K × Ω and for ε > 0, there exist a neighborhood U of x in K and an open subset set G in E with A ⊂ G such that for all (y, B) ∈ K ×Ω with y ∈ U and B ⊂ G, | p(y, B) - p(x, A)| < ε, and (ii) for each fixed X ∈ K, p(x, ·) is a convex function on Ω. If p(x, x − f(x)) ≤ p(x, g(x) − f(x)) for all x ∈ K, and if, for each x ∈ K with f(x) ∩ g(x) = ø, there exists y ∈ K with p(x, y − x + g(x) − f(x)) < p(x, x − f(x)), then there exists an x0 ∈ K such that f(x0) ∩ g(x0) ≠ ø. Another coincidence theorem on a nonempty compact convex subset of a Hausdorff locally convex topological vector space is also given.

scholarly journals On the representation of strictly continuous linear functionals

1981 ◽  
Vol 24 (2) ◽  
pp. 123-130 ◽  
Author(s):  
Liaqat Ali Khan ◽  
K. Rowlands
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Hausdorff Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Completely Regular ◽  
Totally Bounded ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

scholarly journals Superdiagonal forms for completely continuous linear operators

1982 ◽  
Vol 23 (2) ◽  
pp. 163-170 ◽  
Author(s):  
Demetrios Koros
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Complex Field ◽  
Continuous Linear ◽  
Completely Continuous ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex

Altman [1] showed that Riesz-Schauder theory remains valid for a completely continuous linear operator on a locally convex Hausdorflf topological vector space over the complex field. In a later paper [2], he proved an analogue of the Aronszajn-Smith result; specifically, he showed that such an operator possesses a proper closed invariant subspace. The purpose of this paper is to show that Ringrose's theory of superdiagonal forms for compact linear operators [3] can be generalized to the case of a completely continuous linear operator on a locally convex Hausdorff topological vector space over the complex field. However, the proof given in [3] requires considerable modification.

scholarly journals Best approximation theorems for composites of upper semicontinuous maps

1995 ◽  
Vol 51 (2) ◽  
pp. 263-272 ◽  
Author(s):  
Sehie Park
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Best Approximation ◽  
Broad Class ◽  
Compact Convex ◽  
Approximation Theorems ◽  
Upper Semicontinuous ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Convex Subsets

Let (E, τ) be a Hausdorff topological vector space and (X, ω) a weakly compact convex subset of E with the relative weak topology ω. Recently, there have appeared best approximation and fixed point theorems for convex-valued upper semicontinuous maps F: (X, ω) → 2(E, τ) whenever (E, τ) is locally convex. In this paper, these results are extended to a very broad class of multifunctions containing composites of acyclic maps in a topological vector space having sufficiently many linear functionals. Moreover, we also obtain best approximation theorems for classes of multifunctions defined on approximatively compact convex subsets of locally convex Hausdorff topological vector spaces or closed convex subsets of Banach spaces with the Oshman property.

scholarly journals Extended Schauder decompositions of locally convex spaces

1974 ◽  
Vol 15 (2) ◽  
pp. 166-171 ◽  
Author(s):  
J. H. Webb
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Locally Convex Spaces ◽  
Unique Point ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Convex Spaces

Let E[τ] be a locally convex Hausdorff topological vector space. An extended decomposition of E[τ] is a family {Ea}α∈A of closed subspaces of E such that, for each x in E and each α in A, there exists a unique point xα in Eα, with Here convergence will have the following meaning. Let Ф denote the set of all finite subsets of A. The sum is said to be convergent to x if for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф containing φ0. It follows that is Cauchy if and only if, for each neighbourhood U of 0 in E, there is an element φ0 of Ф such that , for all φ in Ф disjoint from φ0.

scholarly journals A class of factorable topological algebras

1990 ◽  
Vol 33 (1) ◽  
pp. 53-59 ◽  
Author(s):  
E. Ansari-Piri
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Approximate Identity ◽  
Necessary Condition ◽  
Topological Algebras ◽  
Approximate Identities ◽  
Cohen Factorization ◽  
Locally Convex ◽  
Space Property ◽  
Uniformly Bounded

The famous Cohen factorization theorem, which says that every Banach algebra with bounded approximate identity factors, has already been generalized to locally convex algebras with what may be termed “uniformly bounded approximate identities”. Here we introduce a new notion, that of fundamentality generalizing both local boundedness and local convexity, and we show that a fundamental Fréchet algebra with uniformly bounded approximate identity factors. Fundamentality is a topological vector space property rather than an algebra property. We exhibit some non-fundamental topological vector space and give a necessary condition for Orlicz space to be fundamental.

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