Orthogonal basis for $L^2[0,1]$

Question

How would I show that $\{e^{2\pi i (n+1/2)t} \}_{n \in \mathbb{Z}}$ is an orthogonal basis for $L^2[0,1]$. Orthogonality is clear. I don't know how to go about proving completeness.

The usual proof for completeness of $\{e^{2\pi i n t}\}_{n \in \mathbb{Z}} $ uses the fact that this set forms an algebra. We don't have this property for the complex exponentials I gave above.

Answer 1

HINT:

Your set is obtained from the standard basis $(e^{2\pi i n t})_{n\in \mathbb{Z}}$ by the transformation $$f(t) \mapsto e^{\pi i t} \cdot f(t)$$ which is an isometry.