Abstract
It is well known that a normed space is uniformly convex (smooth) if and only if its dual space is uniformly smooth (convex). We extend the notions of uniform convexity (smoothness) from normed spaces to countably normed spaces ‘in which there is a countable number of compatible norms’. We get some fundamental links between Lindenstrauss duality formulas. A duality property between uniform convexity and uniform smoothness of countably normed spaces is also given. Moreover, based on the compatibility of those norms, it is interesting to show that ‘from any point in a real uniformly convex complete countably normed space, the nearest point to a nonempty convex closed subset of the space is the same for all norms’, which is helpful in further studies for fixed points.
Highlights
Definition . (Uniformly convex space [ – ]) A normed linear space E is called uniformly convex if for any ∈ (, ] there exists a δ = δ( ) > such that if x, y ∈ E with x =, y = and x – y ≥ (x + y) ≤ – δ.Definition . (Modulus of convexity [ – ]) Let E be a normed linear space with dim E ≥
In Proposition . , we showed the conditions of equivalence to uniform convexity of countably normed spaces, and in Theorem . , we show the conditions of equivalence to uniform smoothness of countably normed spaces
We prove one of the fundamental links between the Lindenstrauss duality formulas
Summary
Let K be a nonempty convex subset of a real normed linear space E. (Metric projection [ , , , ]) Let E be a real uniformly convex and uniformly smooth Banach space, K be a nonempty proper subset of E. By extending some theorems to the case of uniformly convex uniformly smooth countably normed spaces, we prove the existence and uniqueness of nearest points in these spaces.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
