As we know the Hessian matrix is symmetric in a finite-dimensional environment. What about the Hessian operator $D^2F$ for a functional $F:H\rightarrow \mathbb{R}$, where $H$ is a Hilbert space and $D$ is the Fréchet derivative?


More specifically, $D^2F(p)$ ($p$ is a critical point of $F$) plays an important role in infinite-dimensional Morse theory, but little proof of the self-adjointness of this operator is shown in any book I have read about infinite-dimensional Morse theory. I would be really grateful for anyone who can show me more details about this.



Is Hessian operator self-adjoint on infinite dimensional environment?
$d^2F(p)$ is symmetric as a bilinear form. See 5.11 of [here](http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf), for example.


