Definition 30.7.1 (Power Series).label Let $E, F$ be locally convex spaces $K \in \RC$ with $F$ being complete, $\bracsn{T_n}_{0}^{\infty}$ 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.
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