Definition 26.6.1 (Power Series).label Let $E$ be a normed space over $K \in \RC$, $F$ be a Banach space over $K$, $\bracsn{T_n}_{0}^{\infty}$ with $T_{n} \in L^{n}(E; F)$ for each $n \in \natz$, and $a \in E$, then the power series of $\bracsn{T_n}_{0}^{\infty}$ about $a$ is the function
\[f(x) = \sum_{n = 0}^{\infty} T_{n}(x - a)^{(n)}\]
defined on points on which the series converges.