So I have been following a book and understand the following version of Taylor's theorem:
Let $X$ be a Banach space and $Y$ a separable Banach space, $A \subset X$ open and convex, $f\in C^n(A, Y)$ and $x_0 \in A$. Then \begin{align*} f(x_0 + h) &= \sum_{k=0}^n \frac{1}{k!} D^k f(x_0)(h, \dots, h) + o(\|h\|^n)\\ \text{ as } h \rightarrow 0 \end{align*}
Now I would like to know whether or not the following is true:
If for each $x_0 \in A$, there exist bounded multilinear maps $D_k(x_0):X^k \rightarrow Y$ such that $$f(x_0 + h) = \sum_{k=0}^n D_k(x_0)(h, \dots, h) + o(\|h\|^n) \text{ as } h \rightarrow 0$$ then $f \in C^n(A,Y)$ and $D^kf(x_0)(h, \dots, h) = k! D_k(x_0)(h, \dots, h)$
Essentially, I am wondering how to make the follwing observation formal: If $f(x) = \langle x,x \rangle$, then \begin{align*} f(x+h) &= \langle x,x \rangle + + 2\langle x,h \rangle + \langle h,h \rangle\\ &= f(x) + Df(x)(h) + \frac{1}{2}D^2f(x)(h,h) \end{align*}
I'm guessing this is true since Gelfand and Fomin's book takes this as the definition of 2nd order differentiability, but I am pretty stuck on how to show it. Showing Taylor's theorem was hard enough because the proof I saw used a slick calculus trick. Are there any recommended references with this converse?