# Reference Request: Vector valued Taylor formula

It is well known that for analytic function defined on interval $I$ we have $$f(x)=\sum^{\infty}_{k=0}f^{k}(0)\frac{x^{k}}{k!}$$ and for function defined on $I^{n}\rightarrow \mathbb{R}$ we have $$f(x)=\sum^{\infty}_{|\alpha|=0}\frac{1}{|\alpha|! }\partial^{\alpha}f(0)x^{\alpha}$$ where $\alpha$ is the multi-index. However, the notation becomes difficult when I work with $$f:I^{n}\rightarrow \mathbb{R}^{m}$$ because even if I can write out the function in coordinates, I lose some of the global picture how they united in $\mathbb{R}^{m}$. Clearly, the best way to work with this is to introduce the derivative as a matrix valued function in $\mathbb{R}^{mn}$. However, this brings the inconvenience of introducing the Frechet derivative and as an example the total second derivative would be in space $\mathbb{R}^{mn^{2}}$. The Taylor formula now becomes $$f(x)=f(0)+f'(0)\cdot x+\frac{1}{2}(f''(0)(x)\cdot x)\cdots$$ I am sure this must be well known in literature. But I am having trouble to find a nice readable form that I can manipulate the indices to show some basic properties like when is $f$ analytic on $I^{n}$. I found the high dimensional derivative very difficult to manipulate algebraically. Can someone give me a hint how this operates in real life?

For example, for now the "working definition" I used for $F$ being analytic is

$$|\partial^{\alpha}F_i(0)|\le \frac{|\alpha|!K^{|\alpha|+1}}{C(d,|\alpha|)},\forall i,\forall |\alpha|>0$$ where $C(d,|\alpha|)$ is a constant independent of $F(x)$ denoting how many terms we have with degree $|\alpha|$. This should be equivalent to the real one using Frechet derivative using triangular inequality. But I feel this is not the right definition and it is also cumbersome to use.

(This question came from my attempt to prove Cauchy-Kowalevski theorem)

Reference:

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I think Dieudonne's modern analysis is worth a look. – James S. Cook Jun 2 '14 at 20:57