Plancherel's Identity proof - justifying order of integration?

[GregA]

SOLVED

With the assumption $$f(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$$ (in a Lebesque sense) I'm trying to include a short proof of Plancherel's identity into my dissertation but am having trouble justifying the change of integration at the end of the following line:

$$\int_{\mathbb{R}}|f(x)|^2\, \mathrm{d}x =\int_{\mathbb{R}}f(x)\overline{f(x)}\mathrm{d}x=\int_{\mathbb{R}}f(x)\left [\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\overline{{\hat{f}(\xi)e^{ix\xi}}}\mathrm{d}\xi\right ]\mathrm{d}x$$

I am aware of Fubini's theorem but my problem is that I can't see why the integral of $\left |\hat{f}(\xi)\right |$ is necessarily bounded in $$L^1$$ so that I can use it. [EDIT] Am I allowed to say that since I know $$f(x)\in L^2$$ then the equalitities give that $\int f(\xi)\mathrm\,{d}\xi$ is bounded?

A number of texts just swap $$\mathrm{d}x$$ and $$\mathrm{d}\xi$$ without worrying this, whilst others just give a hand wavy argument that it all works out ok.

I know that it can be proved using the convolution theorem but the argument is longer. Am I missing something obvious or would anybody be able to point me in the right direction for resolving this problem? (and I acknowledge I have gaps in my knowledge about function spaces etc...)

 

[mathman]

If f(x) is in L1, its Fourier transform is bounded.