lower semi continuous on Banach space implies locally bounded?

Question

Let $(X, \|\cdot\|)$ be a Banach space; and $f:X\to [0, \infty)$ is lower- semi continuous on $X.$

My Question is: Can we expect $f$ is bounded in some open subset of $X$ ?

[If answer is positive, I guess, may be Baire category theorem is useful; but I don't know how? ]

Answer 1

Yes there is such an open set.

Consider the sets $B_n = \{x \in X : f(x) \le n\}$. By lower-semicontinuity they are closed. If one of them has a non-empty interior, we are done. On the other hand $X = \bigcup_n B_n$, and by the Baire category theorem this is only possible if at least one $B_n$ has a non-empty interior.