Let $V,W$ be nonzero normed spaces over $\mathbb{K}$.
Let $E$ be open in $V$ and $f:E\rightarrow W$ be a twice Fréchet-differentiable function.
Then, $D^2 f: E\rightarrow \mathscr{L}_2(V^2,W)$ is symmetric. That is, at any point $p\in E$, $((D^2 f)(p) )(x,y)= ((D^2 f) (p))(y,x)$.
It is not that I don't understand the proof, but I don't understand why it must hold. What's the geometric meaning of higher order Fréchet derivatives?
First order Fréchet derivative $(D f)(p)$ is a function that approximates $f(x)$ where $x$ near $p$, by means of a linear transformation. From the definition $(D^2 f)(p)$ is a linear approximation of $(Df)(x)$ where $x$ is near $p$. However, I'm not sure what does $D^2 f$ say about $f$.
What does $D^n f$ represent of $f$ exactly?
Thank you in advance.