I'm trying to solve the following problem. Let $f_0 \in L^p(\mathbb{R})$ and let $f_n(x)=f_0(x+n)$. Show that $f_n$ converges weakly to zero in $L^p(\mathbb{R})$.
I know that if $(x_n)$ is a sequence in $X$, then we have the following.
$\forall f \in X^*, f(x_n)$ converges to $f(x)$ iff $(x_n)$ converges to $x$ in the weak topology.
I know the dual space of $L^p$ is $L^q$ where $\frac{1}{p}+\frac{1}{q}=1$.
I've been playing around with some of the well-known properties of $L^p$ spaces but haven't made much progress.
You can argue by density. Given $g\in L^q$, choose a compactly supported $h$ such that $\|g-h\|_q<\epsilon$.
Then you have $$ \int gf(x+n)\,dx=\int h f(x+n)\,dx+\int (g-h) f(x+n)\,dx. $$ The first integral approaches zero as $n\to \infty$ because you are integrating over some bounded interval (the support of $h$) $$ \int h f(x+n)\,dx=\int\limits_{-L}^L h f(x+n)\,dx\le C\int\limits_{-L}^L f(x+n)\,dx= $$ $$ =C\int\limits_{-L+n}^{L+n} f(x)\,dx\le C_1\left[\int\limits_{-L+n}^{L+n} f(x)^p\,dx\right]^{1/p}\to 0\quad\textrm{as} n\to\infty $$ as it is the "tail" of a convergent integral ($C$ is the max of $|h|$ and $C_1$ depends on $L$ here as you are using Holder's inequality).
As for the second integral, it is bounded by $$ \|(g-h)\|_q \|f(x+n)\|_p\le \epsilon\|f\|_p $$ by Holder's inequality. Choosing first $\epsilon$ small and then $n$ large proves your claim.