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.

On Convergence of Projections in Locally Convex Spaces

1966 ◽  
Vol 9 (1) ◽  
pp. 107-110
Author(s):  
J. E. Simpson
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Linear Mapping ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Continuous Linear Mapping ◽  
Basic Assumptions ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

This note is concerned with the extension to locally convex spaces of a theorem of J. Y. Barry [ 1 ]. The basic assumptions are as follows. E is a separated locally convex topological vector space, henceforth assumed to be barreled. E' is its strong dual. For any subset A of E, we denote by w(A) the closure of A in the σ-(E, E')-topology. See [ 2 ] for further information about locally convex spaces. By a projection we shall mean a continuous linear mapping of E into itself which is idempotent.

Schauder decompositions in locally convex spaces

1970 ◽  
Vol 68 (2) ◽  
pp. 377-392 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
General Theory ◽  
Vector Space ◽  
Topological Vector Space ◽  
Locally Convex Spaces ◽  
One Dimensional ◽  
The One ◽  
Locally Convex ◽  
Special Case ◽  
Convex Spaces

A decomposition of a topological vector space E is a sequence of non-trivial subspaces of E such that each x in E can be expressed uniquely in the form , where yi∈Ei for each i. It follows at once that a basis of E corresponds to the decomposition consisting of the one-dimensional subspaces En = lin{xn}; the theory of bases can therefore be regarded as a special case of the general theory of decompositions, and every property of a decomposition may be naturally denned for a basis.

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 few remarks on bounded operators on topological vector spaces

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 763-772
Author(s):  
Omid Zabeti ◽  
Ljubisa Kocinac
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Topological Vector Space ◽  
Compact Operators ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Bounded Operators ◽  
Topological Vector Spaces ◽  
Different Types ◽  
Locally Convex

We give a few observations on different types of bounded operators on a topological vector space X and their relations with compact operators on X. In particular, we investigate when these bounded operators coincide with compact operators. We also consider similar types of bounded bilinear mappings between topological vector spaces. Some properties of tensor product operators between locally convex spaces are established. In the last part of the paper we deal with operators on topological Riesz spaces.

On the Nonstandard Duality Theory of Locally Convex Spaces

1980 ◽  
Vol 32 (2) ◽  
pp. 460-479 ◽  
Author(s):  
Arthur D. Grainger
Keyword(s):  
Vector Space ◽  
Duality Theory ◽  
Dual System ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Nonstandard Model ◽  
Locally Convex ◽  
External Property ◽  
Convex Spaces

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

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.

Schauder bases and decompositions in locally convex spaces

1974 ◽  
Vol 76 (1) ◽  
pp. 145-152 ◽  
Author(s):  
J. H. Webb
Keyword(s):  
Linear Operator ◽  
Vector Space ◽  
Topological Vector Space ◽  
Weak Topology ◽  
Locally Convex Spaces ◽  
Partial Summation ◽  
Schauder Bases ◽  
Summation Operator ◽  
Image Position ◽  
Locally Convex

Let E[τ] be a locally convex Hausdorif topological vector space, with a Schauder basis {xi, x′j wherefor each x ∈ E. The partial summation operator Sn, defined byis a linear operator on E, whose definition extends at once to a linear operator mapping (E′)* into E, where (E′)* is the algebraic dual of E′. The dual of Sn is the operator S′n, mapping E* into E′, defined byand 〈Snx, x′〉 = 〈x, S′nx′〉 for each x ∈ (E′)*. It is easy to see that S′nx′ → x′ with respect to the weak topology σ(E′, E) for each x′ ∈ E′.

Sequential Compactness of X Implies a Completeness Property for C(X)

1976 ◽  
Vol 28 (1) ◽  
pp. 207-210 ◽  
Author(s):  
M. Rajagopalan ◽  
R. F. Wheeler
Keyword(s):  
Hausdorff Space ◽  
Locally Convex Spaces ◽  
Compact Open Topology ◽  
Von Neumann ◽  
Completely Regular ◽  
Totally Bounded ◽  
Hausdorff Topological Vector Space ◽  
Sequential Compactness ◽  
Locally Convex ◽  
Convex Spaces

A locally convex Hausdorff topological vector space is said to be quasicomplete if closed bounded subsets of the space are complete, and von Neumann complete if closed totally bounded subsets are complete (equivalently, compact). Clearly quasi-completeness implies von Neumann completeness, and the converse holds in, for example, metrizable locally convex spaces. In this note we obtain a class of locally convex spaces for which the converse fails. Specifically, let X be a completely regular Hausdorff space, and let CC(X) denote the space of continuous real-valued functions on X, endowed with the compact-open topology.

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.

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.)

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