In this paper, we study briefly the role played by the locally uniformly rotund (LUR) norm and Frechet differentiability of a norm on the Banach space theory. Our old outstanding open Problem 3.8 mentioned below is the main object of this paper. We study nearly about it and find some additional assumptions on the space attached with this problem to obtain its positive or negative answer. We investigate different results related to these norms and their duals on different settings. In particular, we introduce reflexive spaces, Banach spaces with unconditional basis, weakly locally uniformly rotund (WLUR) norm, Almost locally uniformly rotund (ALUR) norm, strongly exposed point, sub-differentiability and ϵ-sub-differentiability, σ–slicely continuity, weakly compactly generated (WCG) Banach spaces with ck –smooth norms, Symulian’s Theorem, and some technical lemmas.
Damai, Gaj R. and Bajracharya, Prakash M.
Frechet Differentiable Norm and Locally Uniformly Rotund Renormings,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 12,
1, Article 18.
Available at: https://digitalcommons.pvamu.edu/aam/vol12/iss1/18