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

  • $\begingroup$ I think that finite dimensional $l^p$ spaces are already a counter-example. The norm is not differentiable at zero. Or am I missing something? $\endgroup$ – Igor Khavkine Jun 12 '14 at 21:50
  • $\begingroup$ @ Dear Benjamin Dickman , thanks, your answer is very perfect $\endgroup$ – user62498 Aug 29 '14 at 7:41

Yes, this is true. See Theorem 8.6 in:

Fabian, M. J. (Ed.). (2001). Functional analysis and infinite-dimensional geometry (Vol. 8). Springer.

enter image description here

(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.)

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.