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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.

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  • $\begingroup$ What is $R$ in the question? $\endgroup$ – Benjamin Nov 10 '15 at 2:09
  • $\begingroup$ $R$ is just the real number field. $\endgroup$ – Peter Nov 10 '15 at 2:23
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$d^2F(p)$ is symmetric as a bilinear form. See 5.11 of here, for example.

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    $\begingroup$ An important thing that Peter Michor didn't say: it is a lot more natural to think of the Hessian as a bilinear form, instead of as an operator. $\endgroup$ – Willie Wong Nov 11 '15 at 15:58

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