Relation between distribution and $L^p$ I'm reading Functional Analysis by Rudin.
We know that every $L^{p},\ 1\leq p \leq\infty$ function can be understood as a (tempered) distribution. (Ah, we're in $\mathbb{R}^n$)
We also know that the dual of $L^{p}$ is isometric to $L^{q}$ unless $p=\infty$.
My question is: for $1\leq p < \infty$ and its conjugate $q$, and $\Lambda\in\mathcal{D}'$, assume $$\sup_{\phi\in\mathcal{D},||\phi||_q\leq1}|\Lambda(q)|<\infty.$$ Can we say that $\Lambda\in L^p$? (i.e. $\exists f\in L^p$ s.t. $f(\phi)=\Lambda(\phi)$ for all $\phi\in\mathcal{D}$.)
If the answer is affirmative, please let me know the proof. External links or PDF files are also welcome.
Thanks!
(p.s. This is my first question in this website. Any comments or tips are appreciated.)
 A: This is correct. To show it, consider a smooth function $\psi$ which is supported in $B(0,1)$, it is positive and has integral $1$, and set $\psi_m(x)=m\psi(mx)$. Consider now the convolutions $\Lambda*\psi_m$, which are functions, and note that, for any $\phi\in\mathcal{D}$ with $\|\phi\|_q\leq 1$, $$\left|\int_{\mathbb R^n}(\Lambda*\psi_m)\cdot\phi\right|=|\Lambda(\tilde{\psi}_m*\phi)|\leq  M,$$ where $M$ is the supremum in your question, and where $\tilde{\psi}$ is the reflection $\tilde{\psi}_m(x)=\psi_m(-x)$, since $\|\tilde{\psi}_m*\phi\|_q\leq 1$.
Suppose now that $p>1$. Then $q<\infty$, therefore $\mathcal{D}$ is dense in $L^q$, and this shows that $$\left|\int_{\mathbb R^n}(\Lambda*\psi_m)\cdot f\right|\leq M$$ for all $f\in L^q$ with $\|f\|_q\leq 1$. Therefore, $(\Lambda*\psi_m)$ is a bounded sequence in $L^p$, hence a subsequence of it converges weakly to a function $f\in L^p$. But, $\Lambda*\psi_m$ converges to $\Lambda$ in the space of distributions, therefore $\Lambda$ can be identified with $f$, which is an $L^p$ function.
If, now $p=1$, then we obtain that $$\left|\int_{\mathbb R^n}(\Lambda*\psi_m)\cdot f\right|\leq M$$ for all $f\in C_0(\mathbb R^n)$ (continuous functions that vanish at $\infty$). Therefore $\Lambda*\psi_m$ is a bounded Borel measure on $\mathbb R^n$, with total variation bounded by $M$. Since a subsequence of $(\Lambda*\psi_m)$ converges weakly in the space of Borel measures, we deduce that $\Lambda$ can be identified with a bounded Borel measure.
