The evaluation functional is defined as: $$A_t(f)=f(t)$$where $A_t : C[a,b] \to \Bbb R$ and $t\in\Bbb R$. One of the other exercises was to show that the same functional is continuous if the space of continuous functions on $[a,b]$ is equipped with the supremum norm, and that proof is clear to me. It is already stated that this does not hold when the same space has the $L^p$ norm. How does one show this?
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It's because there are integrable functions that are still very spiky. Formally, the linear functional $A_t$ is bounded if
$$\sup_{\|f\| = 1} \|A_t(f)\| < \infty$$
where the norms are taken in appropriate spaces. In this case, this is the statement
$$\sup_{\|f\|_p = 1} |f(t)|$$
But if $p < \infty$, a small $L^p$ norm on a function doesn't imply a small bound on the function. I'll let you work out how to come up with counterexamples (think about, say, triangles).