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$L^{2}$

L^{p}

for

p = = 2

.

L^{2}

is by far the most important of

L^{p}

because it is quantum mechanics states live, because the total probability of being in any state has to be 1!

L^{2}

has some crucially important properties that other

L^{p}

don't (TODO confirm and make those more precise):

it is the only $L^{p}$ that is Hilbert space because it is the only one where an inner product compatible with the metric can be defined:
- math.stackexchange.com/questions/2005632/l2-is-the-only-hilbert-space-parallelogram-law-and-particular-ft-gt
- www.quora.com/Why-is-L2-a-Hilbert-space-but-not-Lp-or-higher-where-p-2
Fourier basis is complete for $L^{2}$ , which is great for solving differential equation

Table of contents 215 3
- Plancherel theorem $L^{2}$ 135 2
  - The Fourier transform is a bijection in $L^{2}$ Plancherel theorem 53
  - Every Riemann integrable function is Lebesgue integrable Plancherel theorem 7

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