scholarly journals Extension with larger norm and separation with double support in normed linear spaces

1980 ◽  
Vol 21 (1) ◽  
pp. 93-105 ◽  
Author(s):  
Ivan Singer
Keyword(s):  
Convex Sets ◽  
Separation Theorems ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Linear Spaces ◽  
Double Support ◽  
Linear Subspaces ◽  
Normed Linear Spaces ◽  
Whole Space

We prove, in normed linear spaces, the existence of extensions of continuous linear functionals from linear subspaces to the whole space, with arbitrarily prescribed larger norm. Also, we prove that under an additional boundedness assumption, in the known separation theorems for convex sets, there exist hyperplanes which separate and support both sets.

scholarly journals Hahn-Banach type theorems and the separation of convex sets for fuzzy quasi-normed spaces

2021 ◽  
Vol 7 (3) ◽  
pp. 3290-3302
Author(s):  
Ruini Li ◽  
◽  
Jianrong Wu
Keyword(s):  
Normed Space ◽  
Convex Cone ◽  
Convex Sets ◽  
Separation Theorems ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Normed Spaces ◽  
Separation Of Convex Sets ◽  
Convex Subsets

<abstract> <p>In this paper, we first study continuous linear functionals on a fuzzy quasi-normed space, obtain a characterization of continuous linear functionals, and point out that the set of all continuous linear functionals forms a convex cone and can be equipped with a weak fuzzy quasi-norm. Next, we prove a theorem of Hahn-Banach type and two separation theorems for convex subsets of fuzzy quasinormed spaces.</p> </abstract>

scholarly journals Continuity of Bounded Linear Operators on Normed Linear Spaces

2018 ◽  
Vol 26 (3) ◽  
pp. 231-237
Author(s):  
Kazuhisa Nakasho ◽  
Yuichi Futa ◽  
Yasunari Shidama
Keyword(s):  
Natural Extension ◽  
Linear Operators ◽  
Bilinear Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Linear Spaces ◽  
Lipschitz Continuous ◽  
Normed Linear Spaces ◽  
Whole Space ◽  
System 1

Summary In this article, using the Mizar system [1], [2], we discuss the continuity of bounded linear operators on normed linear spaces. In the first section, it is discussed that bounded linear operators on normed linear spaces are uniformly continuous and Lipschitz continuous. Especially, a bounded linear operator on the dense subset of a complete normed linear space has a unique natural extension over the whole space. In the next section, several basic currying properties are formalized. In the last section, we formalized that continuity of bilinear operator is equivalent to both Lipschitz continuity and local continuity. We referred to [4], [13], and [3] in this formalization.

scholarly journals On strong proximinality in normed linear spaces

2016 ◽  
Vol 70 (1) ◽  
pp. 19
Author(s):  
Sahil Gupta ◽  
T. D. Narang
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Linear Spaces ◽  
Quotient Spaces ◽  
Normed Linear Spaces ◽  
Rotund Banach Space ◽  
Approximative Compactness

The paper deals with strong proximinality in normed linear spaces. It is proved that in  a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and  approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed.

Sign in / Sign up

Export Citation Format

Share Document