Let $X$ be a Banach space. Is it true that if dual norm of $X^*$ is Frechet differentiable then $X$ is reflexive? Can any one help me? thanks
Yes, this is true. See Theorem 8.6 in:
Fabian, M. J. (Ed.). (2001). Functional analysis and infinite-dimensional geometry (Vol. 8). Springer.
(Lemma 8.4 not visible in Google Books.) The citations for the above are:
[Jam2] James, R. C. (1964). Weak compactness and reflexivity. Israel Journal of Mathematics, 2(2), 101-119.
[Dis2] Diestel, J. (1984). Sequences and series in Banach spaces (Vol. 13). Berlin Heidelberg New York: Springer.
Alternatively, see Corollary 2.3 in:
Haghshenas, H. (2009). A Compilation of some Well-Known Results in Renorming Theory. arXiv preprint arXiv:0901.3029.
(Here the author proves the stated fact as a consequence of Smulian’s theorem, which is mentioned in the comments above.)
Other classical results can also be found in:
Diestel, J. (1975). Geometry of Banach spaces: selected topics (Vol. 485). Berlin: Springer-Verlag.
(I think this particular fact is again proved on p. 33.)