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′.

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.

scholarly journals A formula to calculate the spectral radius of a compact linear operator

1997 ◽  
Vol 20 (3) ◽  
pp. 585-588 ◽  
Author(s):  
Fernando Garibay Bonales ◽  
Rigoberto Vera Mendoza
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Compact Operator ◽  
Spectral Radius ◽  
Topological Vector Space ◽  
Closed Subspace ◽  
Minkowski Functional ◽  
Normed Spaces ◽  
Locally Convex

There is a formula (Gelfand's formula) to find the spectral radius of a linear operator defined on a Banach space. That formula does not apply even in normed spaces which are not complete. In this paper we show a formula to find the spectral radius of any linear and compact operatorTdefined on a complete topological vector space, locally convex. We also show an easy way to find a non-trivialT-invariant closed subspace in terms of Minkowski functional.

Banach Envelopes of Non-Locally Convex Spaces

1986 ◽  
Vol 38 (1) ◽  
pp. 65-86 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Bounded Linear Operator ◽  
Locally Convex Spaces ◽  
Minkowski Functional ◽  
Bounded Linear ◽  
Banach Envelope ◽  
Image Position ◽  
Locally Convex

Let X be a quasi-Banach space whose dual X* separates the points of X. Then X* is a Banach space under the normFrom X we can construct the Banach envelope Xc of X by defining for x ∊ X, the normThen Xc is the completion of (X, ‖ ‖c). Alternatively ‖ ‖c is the Minkowski functional of the convex hull of the unit ball. Xc has the property that any bounded linear operator L:X → Z into a Banach space extends with preservation of norm to an operator .

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.

scholarly journals Linear operators whose domain is locally convex

1977 ◽  
Vol 20 (4) ◽  
pp. 293-299 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Vector Space ◽  
Unit Ball ◽  
Topological Vector Space ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Continuous Linear ◽  
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.

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 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 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.

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

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