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I have been trying to solve the following question, without success.

"Find an example of a sequence $(f_n)_{n \in \mathbb{N}}$ of functions in $L_1(\mathbb{R}) \cap L_2(\mathbb{R})$ such that

\begin{equation} \left\lVert f_n \right\rVert_1 \rightarrow 0 \text{, but } \left\lVert f_n \right\rVert_2 \nrightarrow 0 \text{."} \end{equation}

I thought of something like $f_n(x) = \frac{1}{nx^{1/3}}$ for $x \in [-1,1]$ and $f_n(x) = 0$ for $x \notin [-1,1]$, but then $\left\lVert f_n \right\rVert_1 \rightarrow 0$ and $\left\lVert f_n \right\rVert_2 \rightarrow 0$. I can't find a way to keep the $L^2$-norm away from 0 while having the $L^1$-norm going to $0$.

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Hint: Take $$f_n(x) = n\cdot 1_{\left[0,\frac{1}{n^2}\right]}(x)$$

Then $||f_n||_1 = \frac{1}{n}\to 0$ but $||f_n||_2 = 1$

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  • $\begingroup$ I think you mean $\|f_n\|_1=1/n\to0$. $\endgroup$
    – Aweygan
    Dec 22, 2016 at 16:38
  • $\begingroup$ ofc, thx, edited $\endgroup$
    – Gono
    Dec 22, 2016 at 16:39

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