analytic functions from subsets of $\mathbb{R}^p$ This is a question regarding a definition.  What does it mean to be a real analytic function from an open subset $U \subset \mathbb{R}^p$ to a set of $n \times n$ matrices.  I know that on the real line, a function is analytic if it has a convergent power series.  (i.e it can be written as $f(x)=\sum_{n=0}^\infty a_n x^n $.   But what does this mean when $x\in U \subset \mathbb{R}^p$? I came across the following statement in a paper: "Let $L:U\rightarrow G$ be a real analytic function from an open subset $U\subset \mathbb{R}^p$ to G, the set of graph laplacians."    I was unable to come across a more general definition of analytic functions that extended to higher dimensions.  
 A: If $U \subset \mathbb{R}^n$ and $Y$ is a Banach space, then $f: U \rightarrow Y$ is an analytic function if, for each $x \in U$, there exists a sequence $\{c_k \}$ in $Y$ and $a \in \mathbb{R}^n$ such that the quantity
$$ \sum_{k = 0}^{\infty} \sum_{|\alpha| = k} c_k (y - a)^{\alpha}$$
absolutely converges in the norm topology of $Y$ to $f(y)$ for $y$ in an open neighborhood of $x$. Here, $\alpha = (\alpha_1, \ldots, \alpha_n)$ is a multi-index, $|\alpha| := \alpha_1 + \ldots + \alpha_n$, and $(y - a)^n := (y_1 - a_1)^{\alpha_1} \ldots (y_n - a_n)^{\alpha_n}$.
In your case, $Y$ is the set of $n \times n$ matrices, which is indeed a Banach space (since it is finite-dimensional), so the above sum must converge with respect to any matrix norm of choice (since they are all equivalent). Notice that the $c_k$, the coefficients of the power series, are matrix-valued.
Speculation: I bet there is some way one can extend this to $f: U \subset X \rightarrow Y$, where both $X,Y$ are Banach spaces, by saying that $f$ is analytic about $x \in U$ if, for all $y$ in some neighborhood about $x$, there exists a sequence of continuous multilinear maps $L_k: X^{k} \rightarrow Y$ such that
$$ f(y) = f(x) + \sum_{k=1}^{\infty} L_k [(y - x)^{(k)}]$$
converges absolutely, and $(y-x)^{(k)} = (y-x,\ldots, y -x) \in X^k$. 
