Recall a definition of lower semicontinuous function (taken from Wiki).
Suppose $X$ is a topological space, $x_{0}$ is a point in $X$ and $f:X\to \mathbb {R} \cup \{-\infty ,\infty \}$ is an extended real-valued function. We say that $f$ is lower semi-continuous at $x_{0}$ if for every $\epsilon >0,$ there exists a neighborhood $U$ of $x_{0}$ such that $f(x)\geq f(x_{0})-\epsilon $ for all $x$ in $U$ when $f(x_{0})<+\infty $, and $f(x)$ tends to $+\infty$ as $x$ tends towards $x_{0}$ when $f(x_{0})=+\infty$. Equivalently, in the case of a metric space, this can be expressed as $$\liminf _{x\to x_{0}}f(x)\geq f(x_{0}).$$
My question is whether we have a generalization of the definition above to Banach space-valued functions. More precisely,
Let $E$ be a Banach space and $X$ be a topological space. Then we say that $f:X\to E$ is lower semicontinuous at $x_0$ if for every $\epsilon>0,$ there exists a neighbourhood $U$ of $x_0$ such that $\|f(x)\|\geq \|f(x_0)\| -\epsilon$ for all $x\in U$ when $f(x_0)<+\infty.$
Does the definition above make sense?