A Review on Some Geometric Results of the Smulian s Theorem on Frechet Differentiability of Norms

TL;DR

This paper reviews geometric aspects of Smulian's theorem related to Frechet differentiability of norms, exploring its implications for dual spaces and properties like reflexivity and strict convexity.

Contribution

It provides a comprehensive review of Smulian's theorem and its geometric consequences on the differentiability and structure of Banach spaces.

Findings

Proves Smulian's theorem on Frechet differentiability of norms.

Analyzes geometric properties related to Gateaux and Frechet differentiability.

Examines implications for dual space properties such as reflexivity and strict convexity.

Abstract

In this paper, we prove the Smulian s theorem on Frechet differentiability of norm,and present some of its geometric results concerning the Gateaux and Frechet differentiability of norm and properties of the allied space and its dual such as reflexivity and strict convexity.