we define $V:L^2(\mathbb{R})\to L^2(\mathbb{R}) $ by $$Vf(t) = f(t-1).$$ Then I'd like to show that $V$ is an isomorphism
Here's what I've got so far:
$$\left \langle vf(t), vf(t)\right \rangle=\int_{\mathbb{R}}f(t-1)\overline{f(t-1)} dt=\left \langle f(t), f(t)\right \rangle$$
so we have$$\|Vf\|=\|f\|$$ indeed, $V$ is one-to-one and we can easily show that $V$ is linear
How can I prove $V$ is surjective and $V^{-1}$ is continuous?
Thanks in advance.