Suppose that $H$ is a real Hilbert space and that $f:H \to \mathbb{R}$ is differentiable in the Frechet sense. Then we can think of the derivative as a function $f': H \to H^* = H$. Suppose that this function also has a continuous Frechet derivative $f'': H \to B(H)$. My question is now: is it true that $f$ is convex if and only if $f''(x)$ is a positive operator for all $x \in H$, i.e., that $\langle f''(x)y, y \rangle \geq 0$ for all $x,y \in H$?
The question is motivated by the analogous case for $C^2$ functions $f: \mathbb{R} \to \mathbb{R}$, in which case $f$ is convex if and only if $f''(x) \geq 0$ for all $x \in \mathbb{R}$.
Thanks in advance.