Let $E$ and $F$ be two real Banach spaces.
By Schwarz' theorem (or a generalization of it), for $f\in C^2(E,F)$ and $x\in E$, $D^2f(x)$, viewed as a bounded bilinear mapping $E\times E\rightarrow F$, is symmetric.
Is that all we know about the values of the second derivative? In other words, for any symmetric bounded bilinear mapping $\nu:E\times E\rightarrow F$, is there a twice Fréchet differentiable mapping $f:E\rightarrow F$ such that $D^2f(x)=\nu$ for some $x\in E$?
In case it makes a difference, I would be also interested in the special case where $E$ is finite-dimensional.