Fix a function $\varphi \in C_c^\infty(\mathbb{R})$, $\varphi \not\equiv 0$, and set $u_n(x) = \varphi(x + n)$. Let $1 \le p \le \infty$. Does there exist any subsequence $(u_{n_k})$ converging strongly in $L^q(\mathbb{R})$, for any $1 \le q \le \infty$?
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1$\begingroup$ This isn't a research level question and should be posted instead on math.stackexchange.com. It will help to include some context for the problem and an explanation of what you have tried. $\endgroup$– Nate EldredgeCommented Feb 16, 2016 at 22:06
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$\begingroup$ Well, 4 people apparently disagree with me, judging from upvotes, so I'll post the answer. $\endgroup$– Nate EldredgeCommented Feb 17, 2016 at 4:05
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1$\begingroup$ By the way, why do you define $p$ which is never used? $\endgroup$– Nate EldredgeCommented Feb 17, 2016 at 4:07
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No. Since $\varphi$ is compactly supported we have $u_n \to 0$ pointwise. Thus if any subsequence converges in $L^q$ the limit must be 0 (a.e.). But by translation invariance $\|u_n\|_q = \|\varphi\|_q \ne 0$ for all $n$, so no subsequence can converge to 0 in $L^q$.
answered Feb 17, 2016 at 4:07