{id=l2}

<\LP> for $p == 2$.

$\LTwo$ is by far the most important of $\LP$ because it is <mathematical formulation of quantum mechanics>[quantum mechanics states] live, because the total probability of being in any state has to be 1!

<l2> has some crucially important properties that other $\LP$ don't (TODO confirm and make those more precise):
* it is the only $\LP$ that is <Hilbert space> because it is the only one where an inner product compatible with the metric can be defined:
 * https://math.stackexchange.com/questions/2005632/l2-is-the-only-hilbert-space-parallelogram-law-and-particular-ft-gt
 * https://www.quora.com/Why-is-L2-a-Hilbert-space-but-not-Lp-or-higher-where-p-2
* <fourier basis is complete for l2>, which is great for solving <differential equation>

$\LTwo$

L2

 L2

 $L^{2}$