# Thread: a < oo; b<oo => a+b < oo

1. ## a < oo; b<oo => a+b < oo

Hello!

Im not in a hurry, but I dont know where to post this (easy) question.

I know this:

$\displaystyle |\hat{f}(\xi)|^2 := |\int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx, |^2 < \infty$

$\displaystyle |\hat{g}(\xi)|^2 := |\int_{-\infty}^{\infty} g(x)\ e^{- 2\pi i x \xi}\,dx, |^2 < \infty$

(Fourier transform)

Now I want to show that

$\displaystyle |\hat{f}(\xi)|^2 + |\hat{g}(\xi)|^2 < \infty$

I guess this is true, because I think it is obvious, but isn't there a professional proof on this?

What do you think?

Any comments would be much appreciated.

Best regards,
Rapha

2. The sum of two finite numbers is finite. Both $\displaystyle |\hat{f}(\xi)|^2$ and $\displaystyle |\hat{g}(\xi)|^2$ are finite, so $\displaystyle |\hat{f}(\xi)|^2+|\hat{g}(\xi)|^2$ is finite. Perhaps there is a more 'professional' way of doing this, but wording it like this seems sufficient.

3. That's as 'professional' as you need to be! The sum of two finite numbers is defined to be a finite number.

4. Ok