scholarly journals Weak compactness in locally convex spaces

1968 ◽  
Vol 9 (2) ◽  
pp. 123-127 ◽  
Author(s):  
I. Tweddle
Keyword(s):  
Banach Space ◽  
General Result ◽  
Real Banach Space ◽  
Weak Compactness ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Weakly Compact ◽  
Convex Case ◽  
Weakly Closed ◽  
Locally Convex

In [2], R. C. James proved that a weakly closed subset X of a real Banach space is weakly compact if and only if each continuous linear form attains its supremum on X. He also extended the result to the locally convex case, and, in [5], J. D. Pryce gave a simplified proof of the general result that is recorded below for reference in the sequel.

Weak Compactness and Separation

1964 ◽  
Vol 16 ◽  
pp. 204-206 ◽  
Author(s):  
Robert C. James
Keyword(s):  
Banach Space ◽  
Weak Compactness ◽  
The Other ◽  
Weakly Compact ◽  
Linear Spaces ◽  
Topological Linear ◽  
Locally Convex ◽  
Compact Subsets

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.

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.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

scholarly journals On the surjectivity of linear maps on locally convex spaces

1982 ◽  
Vol 32 (2) ◽  
pp. 187-211 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Basic Notion ◽  
Linear Map ◽  
Linear Maps ◽  
Locally Convex ◽  
Do So ◽  
Convex Spaces

AbstractThe aim of this note is to investigate the structure of general surjectivity problem for a continuous linear map between locally convex spaces. We shall do so by using the method introduced in Yamamuro (1980). Its basic notion is that of calibrations which has been introduced in Yamamuro (1975), studied in detail in Yamamuro (1979) and appliced to several problems in Yamamuro (1978) and Yamamuro (1979a).

An Embedding Theorem for Separable Locally Convex Spaces

1971 ◽  
Vol 14 (1) ◽  
pp. 119-120 ◽  
Author(s):  
Robert H. Lohman
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Space ◽  
Separable Banach Space ◽  
Locally Convex Space ◽  
Embedding Theorem ◽  
Vector Spaces ◽  
Locally Convex Spaces ◽  
Topological Vector Spaces ◽  
Locally Convex

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

scholarly journals A note on d-ideals in some near-algebras

1967 ◽  
Vol 7 (2) ◽  
pp. 129-134 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Algebra ◽  
Real Banach Space ◽  
Continuous Linear ◽  
Algebraic Operations

Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.

Projections and embeddings of locally convex operator spaces and their duals

1984 ◽  
Vol 96 (2) ◽  
pp. 321-323 ◽  
Author(s):  
Jan H. Fourie ◽  
William H. Ruckle
Keyword(s):  
Dual Space ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Complemented Subspace ◽  
Necessary And Sufficient ◽  
Locally Convex ◽  
Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

scholarly journals On geometric properties of ⊥BJCϵ symmetric in some types of Banach spaces

2020 ◽  
Vol 1664 (1) ◽  
pp. 012038
Author(s):  
Saied A. Jhonny ◽  
Buthainah A. A. Ahmed
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Sphere ◽  
Real Banach Space ◽  
Convex Banach Space ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Geometric Properties ◽  
Strictly Convex ◽  
Strictly Convex Banach Space

Abstract In this paper, we ⊥ B J C ϵ -orthogonality and explore ⊥ B J C ϵ -symmetricity such as a ⊥ B J C ϵ -left-symmetric ( ⊥ B J C ϵ -right-symmetric) of a vector x in a real Banach space (𝕏, ‖·‖𝕩) and study the relation between a ⊥ B J C ϵ -right-symmetric ( ⊥ B J C ϵ -left-symmetric) in ℐ(x). New results and proofs are include the notion of norm attainment set of a continuous linear functionals on a reflexive and strictly convex Banach space and using these results to characterize a smoothness of a vector in a unit sphere.

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