Let
- $X,Y,E$ be $\mathbb R$-Banach spaces
- $f:E\to X$ and $g:E\to Y$ be twice Fréchet differentiable
What's the second Fréchet derivative of $f\otimes g$?
$f\otimes g$ has to be understood as the mapping $$X\to E\:\hat\otimes_\pi\:F\;,\;\;\;x\mapsto f(x)\otimes g(x)\tag1.$$
My own calculations yielded $${\rm D}(f\otimes g)={\rm D}f\otimes g+f\otimes{\rm D}g\tag2$$ and $${\rm D}^2(f\otimes g)={\rm D}^2f\otimes g+2{\rm D}f\otimes{\rm D}g+f\otimes{\rm D}^2g\tag3.$$ However, with $(2)$ we obtain for the simple example $E=X=Y$ and $f=g=\operatorname{id}_X$ $$({\rm D}^2(f\otimes g)(x)y)z=2y\otimes y\;\;\;\text{for all }x,y,z\in E\tag4$$ which seems to be wrong.