Theorem Fourier Analysis The inner product of the two-dimensional sequences $f(x,y)$ and $g(x,y)$ is equal to the inner product of their Fourier transforms, that is:
$$\sum_{x=-\infty}^{\infty}\sum_{y=-\infty}^{\infty}f(x,y)g^*(x,y)=\dfrac{1}{4\pi^2}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}F(w_x,w_y)G^*(w_x,w_y)\,dw_x\,dw_y.$$
I am trying use a Fourier transform inverse and follow re-arranged the integrals and use the Dirac function. But I don't know Why the integrals have limits $(-\pi,\pi)$.

The set $\{e^{imx}e^{iny},m,n\in\Bbb Z\}$ forms a Hilbert basis of the space $H_1:=L^2((-\pi,\pi)^2)$ with the canonical inner product. Denote $e_{m,n}$ the sequence whose all terms are $0$, except the $(m,n)$-th which is $1$. Then $\{e_{m,n},m,n\in\Bbb Z\}$ form a Hibert basis for $H_2:=\ell^2(\Bbb Z^2)$. 
The two mentioned Hilbert spaces are isometric (say $\iota\colon H_1\to H_2$, with $\iota(e^{imx}e^{iny})=e_{m,n}$), so for each $x\in H_1$, $\lVert \iota(x)\rVert_{H_2}²=\lVert x\rVert_{H_1}^2$ . Such an equality is true for $x\pm iy$ and $x\pm y$, which gives 
$$\langle \iota(x),\iota(y)\rangle_{H_2}=\langle x,y\rangle_{H_1},$$
what is wanted.  
