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We first recall two definitions about strong convexity and Frechet differentiability in normed space.

Let $(X, \|.\|)$ be a normed space and $f:X\rightarrow\mathbb{R}$ be a function.

(a) $f$ is said to be strongly convex if there exists $\gamma>0$ such that $$ f(\lambda x+(1-\lambda)y)\leq\lambda f(x)+(1-\lambda)f(y)-\gamma\frac{\lambda(1-\lambda)}{2}\|x-y\|^2\quad \forall x,y\in X, \forall\lambda\in[0, 1]; $$ (b) $f$ is said to be Frechet differentiable at $x_0\in X$ if there exists a continuous linear functional $A_x:X\rightarrow\mathbb{R}$ such that $$ f(x+h)=f(x)+A_x(h)+0(h), $$ where $$ \lim_{h\rightarrow 0}\frac{0(h)}{\|h\|}=0. $$

When $X$ is a real Hilbert space, it is easy to find a strongly convex and Frechet differentiable $f(x)$ (for example $f(x)=\|x\|^2$). While $X$ is a reflexive Banach space but not an inner product space, it is difficult to construct a class of strongly convex and Frechet differentiable functions. These above observations lead us to the following questions:

Question 1. How to construct a reflexive Banach space but not an inner product space and a class of strongly convex and Frechet differentiable functions on that space?

Question 2. The role of the class strongly convex and Frechet differentiable functions in reflexive Banach space but not an inner product space? (Note that class of convex and Frechet differentiable functions in reflexive Banach space are important in optimization problem, calculus of variation, variational analysis,...).

Question 3. How can we find the references (books, papers,...) related to class strongly convex and Frechet differentiable functions in reflexive Banach space but not an inner product space?

I would like to thank for all comments, helping, guidance.

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Could you please explain what do you mean by "constructing a class of strongly convex and Frechet differentiable functions."? –  Tomek Kania Apr 10 at 21:54
    
No one can help me to solve this question? –  impartialmale Apr 12 at 1:09

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