Frechet derivative of "power function"

Question

Let $H$ be a Hilbert space with norm $\|\cdot\|$, and let $p\geq2.$ Define $F(x):=\|x\|^p,\,\forall x\in H.$

Now I want to calculate the first and second order Frechet derivatives of $F$.

When $p=2$, it is straightforward to check that $F^\prime(x)=2(x,\cdot)$ using the Hilbert structure, and it is reasonable to guess that $F^{\prime\prime}(x)=2$.

But when $p$ is other than $2$, the thing seems very different. I guess that $$F^\prime(x)=p\|x\|^{p-2}(x,\cdot)$$ and $$F^{\prime\prime}(x)=(p-1)p\|x\|^{p-3}(x,\cdot)$$ but I cannot check them.

Any help is appreciated.

Answer 1

You can use the chain rule because $x\mapsto\|x\|^p$ is the composition of the mappings $x\mapsto\|x\|^2$ and $\alpha\mapsto\alpha^{\frac{p}{2}}$.