Uniform Convergence. Interchangeability of Limiting Processes. Examples of Banach Spaces. The Theorem of Arzela-Ascoli

2003 ◽  
pp. 47-60
Author(s):  
Jürgen Jost
Keyword(s):  
Uniform Convergence ◽  
Banach Spaces

Topological decompositions of the duals of locally convex operator spaces

1983 ◽  
Vol 93 (2) ◽  
pp. 307-314 ◽  
Author(s):  
D. J. Fleming ◽  
D. M. Giarrusso
Keyword(s):  
Uniform Convergence ◽  
Banach Spaces ◽  
Linear Subspace ◽  
Locally Convex Spaces ◽  
Continuous Linear ◽  
Operator Spaces ◽  
Natural Topology ◽  
Usual Norm ◽  
Linear Maps ◽  
Locally Convex

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

Two examples concerning uniform convergence of measures w.r.t. balls in Banach spaces

1976 ◽  
pp. 141-146 ◽  
Author(s):  
Flemming Topsøe ◽  
Richard M. Dudley ◽  
Jørgen Hoffmann-Jørgensen
Keyword(s):  
Uniform Convergence ◽  
Banach Spaces

scholarly journals The compact weak topology on a Banach space

1992 ◽  
Vol 120 (3-4) ◽  
pp. 367-379 ◽  
Author(s):  
Manuel González ◽  
Joaquín M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Banach Spaces ◽  
Weak Topology ◽  
Dual Space ◽  
Holomorphic Maps ◽  
Weakly Compact ◽  
Natural Manner ◽  
Locally Convex Topology ◽  
Locally Convex

SynopsisThe compact weak topology (kw) on a Banach space is defined as the finest topology that agrees with the weak topology on weakly compact subsets. It appears in a natural manner in the study of certain classes of continuous and holomorphic maps between Banach spaces. In this paper we treat the kw topology and the finest locally convex topology contained in kw, which we call the ckw topology. We prove that kw = ckw if and only if the space is reflexive or Schur, and we derive characterisations of Banach spaces not containing l1, and of other classes of Banach spaces, in terms of these topologies. We also show that ckw is the topology of uniform convergence on (L)-subsets of the dual space. As a consequence, Banach spaces with the reciprocal Dunford–Pettis property are characterised.

Fredholm Theory in Banach Spaces

1986 ◽  
Author(s):  
Anthony Francis Ruston
Keyword(s):  
Banach Spaces ◽  
Fredholm Theory
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