$X$ is a Banach space. $f$ is a function (may not linear) on $X$.We call $f$ is weakly lower semicontinuous if the set $\{x:f(x)≤c\}$ is weakly sequently closed.We call the set $A$ is weakly sequently closed if any ${x_n}$ belongs to $A$,$x_n$ weakly converge to $x_0$,then $x_0 ∈ A$.
- Show that $f$ is weakly lower semicontinuous iff $x_n ∈ X$, $x_n$ weakly converge to $x_0$ implies $f(x_0)≤$the lower limit of $f(x_n)$.
- $f$ is continuous (may not linear) and convex $$(0≤t≤1,f(tx+(1-t)y)≤tf(x) + (1-t)f(y))$$ imply that $f$ is weakly lower semicontinuous.
