Question on the evaluation functional in $L^p$ I know that for $1 \leq p < \infty$ we have $C_c(X)$ dense in $L^p(X,\mu)$. In $L^p(X,\mu)$ the evaluation functional is not bounded, but in $C_c(X)$ it is.
So if I define $T_x : C_c(X) \to \mathbb{R}$ as $T_x f = f(x)$ for $x \in X$ I can extend this functional to $L^p(X,\mu)$ using the Hahn Banach (HB) theorem.
To make sure I understand how to possibly apply HB I wonder:

*

*Is my application correct?

*Am I right when I say that the extension if $T_x$ is bounded?

Thank you.
 A: You cannot extend $T_x$ to a bounded linear functional via Hahn-Banach because, in general, the evaluation functional in $(C_c(X), ||\cdot||_p)$ is not bounded. For example, let $X = [0,1]$ and let
$$f_n(x) = \begin{cases}
(2n - 2n^2x)^{1/p} & 0 \leq x \leq \frac{1}{n} \\
0 & \text{o.w.}
\end{cases}$$
Observe that $||f_n||_p = 1$ for all $n$. Now let $\delta_0$ be the point evaluation functional at $0$ and observe that: $$\delta_0(f_n) = (2n)^{1/p} \to \infty$$
As $n \to \infty$.
A: You can linearly extend $T_x$ to $L^p$ and you don't even need Hahn-Banach for that. But to extend a linear functional from a subspace $F\subset E$ to a bounded linear functional on $E$, you need it to be continuous on $F$ to begin with - continuous with respect to the norm of $E$, that is. But the functional $T_x$ is in general not continuous with respect to the $L^p$ norm on $C_c(X)$, so there is no chance to extend $T_x$ to a bounded linear functional on $L^p(X,\mu)$.
Of course, things also depend on the measure space. If each point has positive measure, then point evaluation is bounded as pointed out by David C. Ullrich in the comments. Moreover, the measure $\mu$ needs to have full support and give finite mass to compact sets in order to view $C_c(X)$ as a subspace of $L^p(X,\mu)$.
A: Yes, you can use Hahn-Banach to extend this functional to all of $L^p(X,\mu)$. However, since $T_x$ is not bounded, the extended functional is linear, but not continuous. Moreover, the extension is not unique. And for $u \not\in C_c(X)$, $T_x u$ might not be related to $u(x)$.
Putting everything together: Yes, you can do it, but it is not a good idea. Don't try this at home and wear safety goggles ;)
