Counterexamples for L1-convergence not implying L2-convergence and vice versa

Question

I am trying to find counterexamples for the following statements.

Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$.

• $f_n \stackrel{L^1}{\longrightarrow} f$ implies $f_n \stackrel{L^2}{\longrightarrow} f$
• $f_n \stackrel{L^2}{\longrightarrow} f$ implies $f_n \stackrel{L^1}{\longrightarrow} f$

A counterexample for the first statement would be $f_n:= \sqrt{n}\cdot \chi_{[0,1/n]}$ and $f\equiv 0$.

However, I am having trouble figuring out a counterexample for the second statement. I originally thought it was true but apparently it is not... Any hints for a counterexample would be appreciated. I would also like to know if there are any extra conditions that would make the second statement true.

Answer 1

Take $f_n=n^{-1}\chi_{(0,n)}$: then $\lVert f_n\rVert_2=n^{-1/2}$ and $\lVert f_n\rVert_1=1$.