1
$\begingroup$

Smulian lemma says Let $(X, ||.||)$ be a Banach space and$(X^*, ||.||^*)$ and let $x\in S_X=\{x\in X:||x||=1\}$ then

(i) $||.||$ is Frechet diffrentiable at $x$ iff $\lim\limits_{n\to\infty}||f_n-g_n||^*=0$ whenever $f_n, g_n \in S_{X^*}$ satisfy $\lim\limits_{n\to\infty}f_n(x)=\lim\limits_{n\to\infty}g_n(x)=1$ iff $(f_n)\in S_{X^*} $ is convergent whenever $\lim\limits_{n\to\infty}f_n(x)=1$

(ii) $||.||$ is Gateaux diffrentiable at $x$ iff $( f_n-g_n) \stackrel{w^*}\to0$ whenever $f_n, g_n \in S_{X^*}$ satisfy $\lim\limits_{n\to\infty}f_n(x)=\lim\limits_{n\to\infty}g_n(x)=1$

(1) Is it true that we say the norm $||.||$ is Frechet diffrentiable at $x$ if weakly convergence in $X^*$ implies norm convergence in $X^*$?

(2) Is it true that we say the norm $||.||$ is Gateaux diffrentiable at $x$ if weakly convergence in $S_{X^*}$ implies weakly convergence in $X^*$?

Thanks

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.