Proof of Plancherel formula I was looking at this question posted here some time ago. 
How to Prove Plancherel's Formula?
I get it until in the third line he practically says that $\int _{- \infty}^{+\infty} e^{i(\omega - \omega')t} dt= 2 \pi \delta(\omega - \omega')$.
I mean, I would understand if we were integrating over a period of length $2 \pi$, but here the integration is over $\mathbb{R}$. 
P.S. I would have asked this directly to the author of the post, but it's been over a year since he last logged in.

I think he used that $$ 1 = \hat{\delta(w)} $$ so, $$\int _{-\infty}^{+\infty} e^{i(\omega-\omega ')t} dt $$ is the antitransform of $\delta$  values in $(\omega - \omega') $ plus $2\pi$ for definition of antitransform.
