for my class I need to find two sequences of functions such that each satisfies one of the conditions below:
$f_n \in L^p(\Bbb{R})$ such that it converges weakly toward $0$ in $L^p(\Bbb{R})$ but doesn't converge strongly in $L^p(\Bbb{R})$
$f_n \in L^p(\Bbb{R})$ such that it converges almost everywhere toward $0$ in $L^p(\Bbb{R})$ but doesn't converge strongly nor weakly in $L^p(\Bbb{R})$
In both cases, we're using the following notions of strong/weak convergence in $L^p(\Bbb{R})$:
$f_n$ converges strongly toward $f$ in $L^p(\Bbb{R})$ iff $\lim_{n \to \infty} \Vert{f_n - f}\Vert_{L^p}=0$
$f_n$ converges weakly toward $f$ in $L^p(\Bbb{R})$ iff $\forall g \in L^{p'}, \lim_{n \to \infty} \Vert{g*(f_n - f)}\Vert_{L^1}=0$ where $p'$ is defined such that $\frac{1}{p} + \frac{1}{p'} = 1$
Up until now I thought I had found two examples in $f_n(x) = \frac{1}{\sqrt n} \Bbb{1}_{[n,2n]}(x)$ for $1.$ and $f_n(x) = n \Bbb{1}_{[0,\frac{1}{n}]}(x)$ for $2.$ but I just realised both are wrong (or so it seems).
I would appreciate any help to find new examples that work! I'm completely stuck... Thanks in advance