Given a Banach space $X$ we say a sequence in $X$, $\{x_n\}$, converges weakly to $x\in X$ if
$$\lim_n \varphi(x_n)=\varphi(x) \quad \forall \varphi\in X^*.$$
Clearly, if $x_n\rightarrow x$ in $X$ then $x_n$ converges weakly to $x$ since $|\varphi(x_n)-\varphi(x)|\leq \|\varphi\|\|x_n-x\|_X$.
Let $f_n=\chi_{[n,n+1]}$. Show that $f_n$ converges weakly to $0$ in $L^p(\mathbb{R})$ when $1<p<\infty$. Does $f_n \rightarrow 0$ in $L^p(\mathbb{R})$? Does $f_n$ converge weakly to $0$ in $L^1(\mathbb{R})$?
My attempt: I'm confused. I feel like $(\int_{\mathbb{R}} f_n^{p})^{\frac{1}{p}} = 1$ for all $n$, and thus it should also converge weakly to one… Can somebody wakl me through this? I'd really appreciate it…. been stuck on this one for at least a week >.<
Best Answer
Hint: What is $L^1(\mathbb{R})^*$ (Use duality of $L_p$-spaces)?
We have
$$\Vert f_n \Vert_p = \left(\int_\mathbb{R} \chi_{[n,n+1]}^p\right)^{1/p} = 1$$
so indeed, $f_n \not\to 0$ in $L_p$.However, this does not mean that $f_n \to 1$ in $L_p$, since
$$\Vert f_n - 1\Vert_p^p = \int|\chi_{[n,n+1]}-1|^p = +\infty$$