Inner product of functions is preserved by inner product of fourier co-efficients, more to plancherel theorem

Question

By Plancherel theorem the map $T:L^{2}(\mathbf{T}) \longrightarrow l^2(\mathbf{Z})$ defined by $T(f) =(f^{\wedge}(n))_{n \in \mathbf{z}}$ is a surjective isometry.

But I have to show a bit more.That is $\forall f,g \in L^2(T)$ we have to show $\left<f,g\right>_{L^2(\mathbf{T})}=\left<f ^\wedge,g^\wedge\right>_{l^2(\mathbf{Z})}$

where,

$f^\wedge=(f^{\wedge}(n))_{n \in \mathbf{z}}$ and

$g^\wedge=(g^{\wedge}(n))_{n \in \mathbf{z}}$.

Thanks for your help.

Answer 1

Hint:

Consider the expansions of $\langle f+g,f+g \rangle$ and $\langle f+ig,f+ig \rangle$