scholarly journals New directions in convex analysis: the differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 41 (2) ◽  
pp. 333-335
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Linear Spaces ◽  
New Directions ◽  
Topological Linear

scholarly journals The differentiability of convex functions on topological linear spaces

1990 ◽  
Vol 42 (2) ◽  
pp. 201-213 ◽  
Author(s):  
Bernice Sharp
Keyword(s):  
Convex Functions ◽  
Inductive Limit ◽  
Asplund Space ◽  
Fréchet Spaces ◽  
Linear Spaces ◽  
Asplund Spaces ◽  
Continuous Convex Function ◽  
Mazur’S Theorem ◽  
Topological Linear ◽  
Differentiability Properties

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

scholarly journals The Engulfing Property from a Convex Analysis Viewpoint

2020 ◽  
Vol 187 (2) ◽  
pp. 408-420 ◽  
Author(s):  
Andrea Calogero ◽  
Rita Pini
Keyword(s):  
Simple Proof ◽  
Convex Functions ◽  
Convex Analysis ◽  
Strict Convexity

Abstract In this note, we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity are conditions intrinsic to the engulfing property.

scholarly journals INEQUALITIES IN TERMS OF THE GÂTEAUX DERIVATIVES FOR CONVEX FUNCTIONS ON LINEAR SPACES WITH APPLICATIONS

2011 ◽  
Vol 83 (3) ◽  
pp. 500-517 ◽  
Author(s):  
S. S. DRAGOMIR
Keyword(s):  
Convex Functions ◽  
Jensen’S Inequality ◽  
Jensen's Inequality ◽  
Linear Spaces ◽  
Divergence Measures ◽  
Gateaux Derivatives ◽  
Gâteaux Derivatives

AbstractSome inequalities in terms of the Gâteaux derivatives related to Jensen’s inequality for convex functions defined on linear spaces are given. Applications for norms, mean f-deviations and f-divergence measures are provided as well.

Fuzzifying topological linear spaces

2004 ◽  
Vol 147 (2) ◽  
pp. 249-272 ◽  
Author(s):  
Daowen Qiu
Keyword(s):  
Linear Spaces ◽  
Topological Linear
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