About convergent sequence $f_n \to f$ in $L^p(U)$ ( Convergence in norm, passage of limit under integral etc.. ; Evans’s PDE )

lebesgue-integralpartial differential equationsreal-analysis

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 ) :

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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 )

Theorem 1. Let $E$ be a measurable set, $1\le p < \infty$, and $q$ the conjugate of $p$. If $f$ belongs to $L^p(E)$ and $g$ belongs to $L^{q}(E)$, then their product $f\cdot g$ is integrable over $E$ and $$ \int_E|f\cdot g| \le \|f\|_{p} \cdot \|g\|_q.$$

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.

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