Properties of uniform convergent series

Question

I know that if a sequence $f_n$ of functions converges uniformly to a limit function $f$, and $f_n$ is continuous for each $n$, then $f$ is continuous.

I am unsure how this proposition works with series.

Let $f_n:A \rightarrow V$,where A is a set and $V$ is a normed vector space. If $\sum f_n$ is a series and the partial sums $\sum_{i=1}^n f_i$ converge uniformly to $f$, then the series converges uniformly to $f$.

Do the partial sums, or the individual functions $f_n$, have to be continuous for $f$ to be continuous?

or does it not matter which condition I use since the sum of continuous functions are continuous in normed vector spaces? (The converse isn't true though. So requiring the individual functions to be continuous is a stronger condition. )

Answer 1

It turns out that "each $f_n$ is continuous" and "each partial sum $\sum_{k=1}^n f_k$ is continuous" are equivalent statements. You've pointed out that the first statement implies the second. For the converse, suppose that each partial sum $\sum_{k=1}^n f_k$ is continuous, and let $m\ge1$ be arbitrary; we need to prove that $f_m$ is continuous.

• If $m=1$, then $f_1$ is equal to first partial sum $\sum_{k=1}^1 f_k$, which is continuous by assumption.
• If $m>1$, then $f_m$ is the difference of the two partial sums $\sum_{k=1}^m f_k$ and $\sum_{k=1}^{m-1} f_k$, both of which are continuous by assumption.