{c}

$\LP$ is:
* <complete metric space>[complete] under the Lebesgue integral, this result is may be called the <Riesz-Fischer theorem>
* not complete under the <Riemann integral>: https://math.stackexchange.com/questions/397369/space-of-riemann-integrable-functions-not-complete

And then this is why <quantum mechanics> basically lives in <l2>: not being complete makes no sense physically, it would mean that you can get closer and closer to states that don't exist!

TODO intuition

Lebesgue integral of $\LP$ is complete but Riemann isn't

Lebesgue integral of Lp is complete but Riemann isn't

 Lebesgue integral of Lp is complete but Riemann isn't

 Lebesgue integral of $L^{p}$ is complete but Riemann isn't