# Weak Convergence in Lp space

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.