# Is there a text introducing “high order Fréchet derivative” well?

Let $X,Y$ be Banach spaces and $U$ be open in $X$. High-order Fréchet derivatives are defined inductively so that the n-th Fréchet-derivative of a function $F$ is $F^{(n)}:U\rightarrow L(X,L(X,....,L(X,Y))))..))$.

I'm now reading "Giovanni-A primer of nonlinear analysis" and this text says that we can view $L(X,L(X,...,L(X,Y))))..))$ as the space of multilinear maps $L_n(X^n,Y)$, but this text does not say how the space of multilinear maps can be a "normed space" (which norm?) in general. I want to know if there is a text introducing these.

Also, the general expression of the Fréchet-derivative of the composition of Fréchet differentiable functions doesn't look simple ( $D(g\circ f)(p)=Dg(f(p)) \circ Df(p)$ for each point $p$, hence the general function $D(g\circ f)$ cannot be expressed in basic operations). I have seen articles resolve this trouble by introducing tensor products, so that we view $L(X,L(X,....,L(X,Y))).....))$ as $L(\otimes_{i=1}^n X, Y)$. However, those articles also do not say how the tensor products can be a "normed space". (What norm is the most natural one?) I want to know this in general too.

What would be texts introducing these rigorously? I hope someone recommends me mutiple books. Thank you in advance :)

• A nice textbook which introduces higher order Fréchet derivatives is "Differential Calculus" by Cartan. – gerw Aug 20 '15 at 13:01
• @gerw This text is outstanding! Thank you so much! – Rubertos Aug 20 '15 at 17:34
• Since you are satisfied, I will post it as an answer. – gerw Aug 20 '15 at 19:04
• Please check. The author is not Giovanni, but Ambrosetti and Prodi. – Siminore Aug 20 '15 at 19:07