Let $f:X\rightarrow [0,\infty)$ be (resp. weakly) lower semi-continuous on the reflexive Banach space $X$. Let $\ell^p(X)$ denote the space of $p$-summable sequences in $X$, i.e.: $\sum\_{n=1}^{\infty}\|x\_n\|\_x^p<\infty$; here $1\leq p<\infty$. Then, is the "induced" map:
$$
F:(x\_n)\mapsto \sum\_{n=1}^{\infty} f(x\_n),
$$
(resp. weakly) lower semi-continuous? I assume that there exists at-least one $p$-summable sequence for which $F$ is finite-valued.



Lower semi-continuity of induced function on sequences
Let $x = (x\_n) \in \ell^p(X)$ and $F\_N(x) := \sum\_{n=1}^N f(x\_n)$, $N \in \mathbb{N}$. First if each $F\_N$ is l.s.c. (weakly or not), then $F = \sup\_{N \in \mathbb{N}} F\_N$ is l.s.c., second if each $x \to f(x\_n)$ is l.s.c., then $F\_N$ is l.s.c. as a sum of finite many l.s.c. functions and finally $x \to x\_n$ is even continuous. (See f.i. Bourbaki (1989), General Topology IV.6.2). Hence your function $F$ is l.s.c. even without the assumption of $F(x)$ being finite at some point $x$.


