# Properties of the square norm in Banach spaces

Let $X$ be a Banach space with its dual $X^*$. Consider the mapping $f: X\rightarrow \mathbb{R}$ given by $$f(x)=\frac{1}{2}\|x\|^2.$$ We have know that when $X$ is a real Hilbert space ($X=X^*$) then $f$ is strongly convex with modulus $\lambda=1$, i.e. $$\alpha f(x)+(1-\alpha)f(y)\geq f(\alpha x+(1-\alpha)y)+\frac{\alpha(1-\alpha)}{2}\|x-y\|^2$$ for all $x, y\in X$ and $\lambda\in [0, 1]$. Moreover, $f$ is Frechet differentiable and $$\nabla_F f(x)=x\quad \forall x\in X.$$ I do not know besides Hilbert space, what kind of Banach spaces do we have two above properties (Frechet differentiability and strong convexity) of the function $f(x)$.

I would like to thank for all constructive comments, helping and pointing out the references related to this problem.

Note.

• If $X^*$ is strictly convex then $f(x)=\frac{1}{2}\|x\|^2$ is Gateaux differentiable and $$\nabla_G f(x)=\{x^*\in X^*: \|x^*\|^2=\|x\|^2=\langle x^*, x\rangle\};$$
• If $f(x)$ is strongly convex with constant $\lambda=1$ then $X$ is a Hilbert space.
• No one can help me to solve this question? – impartialmale Apr 12 '14 at 1:08
• I do not see your question ?! – user119228 Apr 12 '14 at 9:18

The spaces with Fréchet differentiable norm are usually called Fréchet smooth spaces... which of course is just a name. Šmulian gave a useful characterization of this property: it holds if and only if for every unit vector $x$ and every sequence of unit functionals $f_n$ such that $f_n(x)\to 1$, the sequence $\{f_n\}$ is norm-convergent.
Examples of the spaces that satisfy both properties: $L^p$ and $\ell^p$ for $1<p\le 2$. (For $2<p<\infty$, the space is uniformly smooth and uniformly convex, but the modulus of convexity has power type $p$, not $2$.)