Let $f \in \mathcal{H}(U)$ a holomorphic function, where $U\subset X$ is an open balanced set in an infinite dimensional Banach space $X$, with power series around $0$ $$f=\sum_{n=0}^\infty P_n,$$ and let $K\subset U$ be a compact set. Suppose that $$\sum_{n=0}^\infty \|P_n\|_K < \delta.$$ I want to show that there is a $\varepsilon>0$ such that $$\sum_{n=0}^\infty \|P_n\|_{K+B_\varepsilon}<\delta.$$ This problem is connected with the next question: Seminorms ported by a compact .Here, I showed that the series is always finite.
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