Consider the set $BC([0,\infty))$ of bounded continuous functions on $[0,\infty)$ with supremum norm.
I have shown that there exists a bounded linear functional $F$ on this space such that $F(f)=\lim_{x\rightarrow \infty} f(x)$ if the limit exists and $F(f) \leq \limsup_{x \rightarrow \infty} f(x)$ for all $f \in BC([0,\infty))$.
I have the following question: Is it necessarily true that $F(f) \geq \liminf_{x \rightarrow \infty} f(x)$ for all $f \in BC([0,\infty))$?
If not, can we find a counterexample?
2026-03-26 03:12:22.1774494742
application of Hahn-Banach to space of functions
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