Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 2 / a

The extended-real functional $f:X\to \overline{\mathbb{R}}$ is sequentially lower semicontinuous in the topology ${\tau }_X$ when every sequence $x_n\to x$ in ${\tau }_X$ satisfies

The assertion is understood for every $\eta \in \mathbb{R}$ and for a topology in which sequentially closed sets are closed, in particular the norm topology of a Banach space. Suppose first that $f$ is sequentially lower semicontinuous and $x_n\in {\Omega }_{\eta }=\{x:f(x)\le \eta \}$ converges to $x$. Thenso $x\in {\Omega }_{\eta }$ and the sublevel set is closed. Conversely, if all sublevel sets are closed but lower semicontinuity fails, there are $x_n\to x$ and a real $\eta <f(x)$ with a subsequence satisfying $f(x_{n_k})\le \eta$. Closedness of ${\Omega }_{\eta }$ would put $x$ in that set, a contradiction. This proves the closed-sublevel-set characterization of lower semicontinuity.

Use the extended-real characteristic functionalFor $\eta <0$ its sublevel set is empty, while for every finite $\eta \ge 0$ its sublevel set is $X^{\prime }$. Both are closed when $X^{\prime }$ is closed, so part ii proves that this indicator functional of a constraint set is lower semicontinuous.