scholarly journals λ-similar bases

1987 ◽  
Vol 10 (2) ◽  
pp. 227-232
Author(s):  
Manjul Gupta ◽  
P. K. Kamthan
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Arbitrary Sequence ◽  
The Family ◽  
Schauder Bases ◽  
Continuous Seminorms ◽  
Relationship Of ◽  
Locally Convex

Corresponding to an arbitrary sequence spaceλ, a sequence{xn}in a locally convex space (l.c.s.)(X,T)is said to beλ-similar to a sequence{yn}in another l.c.s.(Y,S)if for an arbitrary sequence{αn}of scalars,{αn p(xn)} ϵ λfor allp ϵ DT⇔{αn q(yn)} ϵ λ, for allq ϵ DS, whereDTandDSare respectively the family of allTandScontinuous seminorms generatingTandS.In this note we investigate conditions onλand the spaces(X,T)and(Y,S)which ultimately help to characterizeλsimilarity between two Schauder bases. We also determine relationship of this concept withλ-bases.

Some criteria for nuclearity

1986 ◽  
Vol 100 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. Sofi
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequence Spaces ◽  
Bilinear Forms ◽  
Simple Formulation ◽  
Normal Topology ◽  
Nuclear Spaces ◽  
Locally Convex

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

scholarly journals The Radon-Nikodym Theorem for Multimeasures

1978 ◽  
Vol 21 (2) ◽  
pp. 167-173
Author(s):  
Le van Tu
Keyword(s):  
Convex Space ◽  
Measurable Space ◽  
Locally Convex Space ◽  
The Family ◽  
Disjoint Sets ◽  
Locally Convex ◽  
Nikodym Theorem

Let (S, ℳ) be ameasurable space(that is, a setSin which is defined a σ-algebra ℳ of subsets) andXa locally convex space. A mapMfrom ℳ to the family of all non-empty subsets ofXis called a multimeasure iff for every sequence of disjoint setsAnɛ ℳ (n=1,2,… )withthe seriesconverges (in the sense of (6), p. 3) toM(A).

The largest family of subsets satisfying sequential-evaluation convergence

2012 ◽  
Vol 49 (3) ◽  
pp. 315-325
Author(s):  
Aihong Chen ◽  
Ronglu Li
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Sequence Space ◽  
Convex Space ◽  
Locally Convex Space ◽  
Sequential Evaluation ◽  
Locally Convex

Suppose X is a locally convex space, Y is a topological vector space and λ(X)βY is the β-dual of some X valued sequence space λ(X). When λ(X) is c0(X) or l∞(X), we have found the largest M ⊂ 2λ(X) for which (Aj) ∈ λ(X)βY if and only if Σ j=1∞Aj(xj) converges uniformly with respect to (xj) in any M ∈ M. Also, a remark is given when λ(X) is lp(X) for 0 < p < + ∞.

Fundamental Biorthogonal Sequences and K-Norms on ϕ

1971 ◽  
Vol 23 (6) ◽  
pp. 1040-1050 ◽  
Author(s):  
L. Crone ◽  
D. J. Fleming ◽  
P. Jessup
Keyword(s):  
Sequence Space ◽  
Convex Space ◽  
Double Sequence ◽  
Locally Convex Space ◽  
Locally Convex

A biorthogonal sequence is a double sequence (xi,fi) where each xi is from some locally convex space X, each fi is from X* and fi(xj) = δij. A biorthogonal sequence is called total if the functionals (fi) are total over X and is called fundamental if sp(xi) is dense in X. If a biorthogonal sequence is both total and fundamental we refer to it as a Markushivich basis or, more simply, an M-basis.If (xi,fi) is a total biorthogonal sequence for X, then X can be identified with the space of all scalar sequences (fi(x)) under the correspondence x ↔ (fi(x)). We refer to this space as the associated sequence space with respect to (xi, fi). With this correspondence, xi corresponds to and fi corresponds to Ei, the ith coordinate functional.

scholarly journals On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
I. Akbarbaglu ◽  
S. Maghsoudi
Keyword(s):  
Convex Space ◽  
Topological Algebra ◽  
Locally Convex Space ◽  
Locally Compact Group ◽  
Compact Groups ◽  
Topological Properties ◽  
Locally Compact ◽  
The Family ◽  
Locally Convex Topology ◽  
Locally Convex

Let be a locally compact group with a fixed left Haar measure and be a system of weights on . In this paper, we deal with locally convex space equipped with the locally convex topology generated by the family of norms . We study various algebraic and topological properties of the locally convex space . In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

scholarly journals The countable neighbourhood property and tensor products

1985 ◽  
Vol 28 (2) ◽  
pp. 207-215 ◽  
Author(s):  
José Bonet
Keyword(s):  
Convex Hull ◽  
Convex Space ◽  
Tensor Products ◽  
Locally Convex Space ◽  
Locally Convex Spaces ◽  
Linear Hull ◽  
Continuous Seminorms ◽  
The Stability ◽  
Locally Convex ◽  
Convex Spaces

This article is intended to enlarge the study of spaces satisfying the countable neighbourhood property and to clarify the incidence of this property in the stability of some locally convex properties of tensor products.We shall use the standard notations of locally convex spaces as in [17] and [18]. The word space will always mean separated locally convex space. If (£, t) is a space, the set of all continuous seminorms on it will be denoted by cs(E). The linear hull and the absolutely convex hull of a subset C of a space will be written lin(C) and г(C) respectively.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

scholarly journals Strictly barrelled disks in inductive limits of quasi-(LB)-spaces

1996 ◽  
Vol 19 (4) ◽  
pp. 727-732
Author(s):  
Carlos Bosch ◽  
Thomas E. Gilsdorf
Keyword(s):  
Convex Space ◽  
Inductive Limit ◽  
Linear Span ◽  
Locally Convex Space ◽  
Minkowski Functional ◽  
Closed Graph ◽  
Inductive Limits ◽  
Locally Convex ◽  
Barrelled Space

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.

scholarly journals In what spaces is every closed normal cone regular?

1970 ◽  
Vol 17 (2) ◽  
pp. 121-125 ◽  
Author(s):  
C. W. McArthur
Keyword(s):  
Banach Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Normal Cone ◽  
Closed Subspace ◽  
Locally Convex Space ◽  
Regular Part ◽  
Fréchet Space ◽  
Sequential Completeness ◽  
Locally Convex

It is known (13, p. 92) that each closed normal cone in a weakly sequentially complete locally convex space is regular and fully regular. Part of the main theorem of this paper shows that a certain amount of weak sequential completeness is necessary in order that each closed normal cone be regular. Specifically, it is shown that each closed normal cone in a Fréchet space is regular if and only if each closed subspace with an unconditional basis is weakly sequentially complete. If E is a strongly separable conjugate of a Banach space it is shown that each closed normal cone in E is fully regular. If E is a Banach space with an unconditional basis it is shown that each closed normal cone in E is fully regular if and only if E is the conjugate of a Banach space.

scholarly journals Extending Edgar's ordering to locally convex spaces

1992 ◽  
Vol 34 (2) ◽  
pp. 175-188
Author(s):  
Neill Robertson
Keyword(s):  
Convex Space ◽  
Dual Pair ◽  
Locally Convex Space ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Hausdorff Topological Vector Space ◽  
Locally Convex ◽  
Mackey Topologies

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

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