Let $U$ be a bounded, connected, open subset of $\mathbb{R}^n$. Assume $1 \le p \le \infty$ .Let $f_n \to f$ be a convergent sequence in $L^p(U)$.
My question is, then,
Q.1. $ \lim_{n\rightarrow\infty} \left\Vert f_n \right\Vert_p =\left\Vert f\right\Vert_p<\infty $?
I found assoicated post : Convergence in Lp implies convergence in Lp norms finite. From the post, I think that the norm convergence is true for $1 \le p < \infty$. And, I also wonder whether this is true for $p=\infty$.
Q.2. For any test function $\phi\in C^{\infty}_c(U)$,
$$ \int_U f \phi dx = \lim_{n\to \infty}\int_U f_n \phi dx \tag{1}$$
? Let's define a functional $T_{\phi}$ on $L^p(U)$ by $T_{\phi}(f):=\int_U f \phi dx $ for all $f\in L^p(U)$. If $T$ is a 'bounded' linear functional on $L^p(U)$, then $T(f_n)$ converges to $T(f)$. But the boundness of $T_{\phi}$ ( including the case $p=\infty$ ) is true? Or is there any other route to show the $(1)$?
This question originates from following proof of the Poincare's inequality in the Evans's PDE Book ( p.275 ) :
Why the underlined statement is true?
Best Answer
For the question 1), this is true by the inverse triangle inequality for norm ; $$\big|\,\|x\|-\|y\|\,\big|\le\|x-y\|.$$
For the question 2), Note next theorem ( Hölder's Inequality )
Now as suggested in my original question, let me show that $T_{\phi}(f) :=\int_U f \phi dx $ ( $f \in L^{p}(U) $) is 'bounded' linear functional so that we are done.
To show the boundness of $T_{\phi}$, we need to show that there exists $M \ge 0$ such that $$ |T_{\phi}(f)| := \bigg| \int_U f \phi dx\bigg| \le M \cdot \|f\|_p $$ for all $f\in L^{p}(U)$.
First note that since $\phi \in C^{\infty}_c(U)$, $\phi \in L^{q}(U)$ for all $1 \le q \le \infty$.
Case 1) $1 \le p < \infty$ : Note that $$\bigg| \int_U f \phi dx \bigg| \le \int_U |f\phi|dx \le \|f\|_p \cdot \|\phi\|_q = \|\phi\|_q \cdot\|f\|_p ,$$ by the Hölder's Inequality
Case 2) $p=\infty$ : In this case, note that
$$ \bigg| \int_U f \phi dx \bigg| \le \int_U |f\phi|dx = \int_U |\phi f |dx \le \|\phi\|_1 \cdot \|f\|_{\infty} $$
also by the Hölder's Inequality.
So we are done. I would like to thank PNDas for his hint ( usage of the Hölder's Inequality ) through comment.