Weak topologies on Banach Spaces

One of the subjects of functional analysis is classification of Banach spaces depending on various properties of the unit ball. The need of such considerations comes from a number of applications to problems of mathematical analysis. The list of subjects contains: differential calculus in normed spaces, approximation theory, weak topologies and reflexivity, general theory of convexity and convex functions, metric fixed point theory, and others. The aim of this book is to present basic facts from this field. It is addressed to advanced undergraduate and graduate students interested in the subject. For some it may result in further interest, a continuation and deepening of their study of the subject. It may be also useful for instructors running courses on functional analysis, supervising diploma theses or essays on various levels.

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Fractional differential inclusions of Hilfer type under weak topologies in Banach spaces

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500151 ◽

2018 ◽

Vol 13 (01) ◽

pp. 2050015

Author(s):

Saïd Abbas ◽

Mouffak Benchohra ◽

Mohamed Abdalla Darwish

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Banach Spaces ◽

Differential Inclusions ◽

Fractional Differential Inclusions ◽

Measure Of Weak Noncompactness ◽

Existence Of Weak Solutions ◽

Mönch’S Fixed Point Theorem ◽

Fractional Differential ◽

Weak Topologies

In this paper, we present some results concerning the existence of weak solutions for some functional Hilfer and Hadamard fractional differential inclusions. The Mönch’s fixed point theorem and the concept of measure of weak noncompactness are the main tools used to carry out our results.

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Lattice bounding, Radonifying and summing mappings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051148 ◽

1975 ◽

Vol 77 (2) ◽

pp. 327-333 ◽

Cited By ~ 8

Author(s):

D. J. H. Garling

Keyword(s):

Banach Spaces ◽

Image Space ◽

Sufficient Condition ◽

Topological Properties ◽

Weak Topologies

There are several known results concerning Radonifying and summing properties of linear mappings in terms of their transposed operators. Thus Schwartz has shown that a sufficient condition for a mapping to be p-Radonifying is that its transposed mapping should be p-decomposing, and that this condition is also necessary if the image space has good topological properties. But as Schwartz observes, these properties are seldom observed by biduals of Banach spaces with their weak-topologies, which arise when 0 ≤ p ≤ 1. This suggests that it would be worth looking for a weaker condition than ‘p-decomposing’.

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