# Fr\'echet-valued formal power series

## TL;DR

This paper establishes conditions under which formal power series with values in a Fréchet space converge in a neighborhood, based on their convergence along complex lines through a non-projectively-pluripolar set.

## Contribution

It provides new sufficient conditions for the convergence of Fréchet-valued formal power series based on line-wise convergence in complex analysis.

## Key findings

- Convergence on a zero-neighborhood is guaranteed under certain conditions.
- Line-wise convergence implies neighborhood convergence in Fréchet spaces.
- The set A's non-projectively-pluripolar property is crucial for the results.

## Abstract

Let $A$ be a non-projectively-pluripolar set in a Fr\'{e}chet space $E.$ We give sufficient conditions to ensure the convergence on some zero-neighbourhood in $E$ of a (sequence of) formal power series of Fr\'{e}chet-valued continuous homogeneous polynomials provided that the convergence holds at a zero-neighbourhood of each complex line $\ell_a := \C a$ for every $a \in A.$