REMARKS:

If f belongs to E', then f is linear. Consequently f(x)-f(a)-f(x-a)=o for all x,a in E. Thus Df(a)=f for all a in E. So f belongs to Da(E) for all a.

We obtained that Da(E) contains E', thus E'=Da(E).

In this context, my proposed request must be modified, to clarify the request, which is not about the trivial case, presented above. I will rephrase the question!

Can we find an infinite dimensional real(or complex) Banach space E, so that it exists a subset A, contained in Ca1(E) - E* ( for some a belongs to E) and the set {Df(a)/f belongs to A} be dense in E'?

Notations: E* is the algebraic dual of E and E' is the continuous dual of E.

Therefore we search differentials of non linear functions from E to R! Also, I think we must search a Banach space with separable dual.

I apologize to all readers for my unclear initial question!

The modification does not save the situation, because one can consider functions that are equal to linear functionals on a fixed ball centered at point a. Even worse, for every fixed Frechet differentiable at the point a function g, the set A = {g + f : f \in E*} has the required property.

Vladimir Kadets

Yes, you have right! Thanks for your answer, professor Kadets!

Please help me solve the following: Let p,q>0 with 1/p+1/q=1. As we know, the continuous dual of Lp identifies with Lq. In the particular case of LRp[0,1], in above context, how can I find a base for LRq[0,1] composed only of differentials of nonlinear functions from LRp[0,1] to R?

Dinu Teodorescu

What do you mean, when speaking about "base for L^q[0,1]"? Do you mean a Schauder basis of a Banach space? If so, then the standard basis is formed by the Haar system. https://encyclopediaofmath.org/wiki/Haar_system

The part about "differentials of nonlinear functions" is inessential, because every element of L^q[0,1] is such a differential.

Vladimir Kadets

Yes, I mean to a Schauder basis in case of separable Banach spaces. Thanks for the link! In the general case I mean to a linear independent vector system generating the space(algebraically) and topologically dense.

In case of Hilbert spaces all is clear for me, but in case of non-separable non-hlibertian Banach spaces, how can we find effectively such systems, generally or only for some particular spaces?

Also, please help! Every element of continuous dual is a "differential of a nonlinear function"? It's generally true? Why? Or only for Lp - spaces?

Many thanks for all explanations!

With all my best regards!

Dinu Teodorescu

For every $f \in X^*$ and every $a \in X$ consider the following $F: X \to R$: $F(x) = f(x) + (f(x-a))^2$. The derivative of $F$ at point $a$ is equal to $f$.

This answers your question "Every element of continuous dual is a "differential of a nonlinear function"? It's generally true? Why?".

Vladimir Kadets

All clear now! Thanks again!