Understanding L^p Spaces: Origins and Applications

[E01]

Can anyone give me a little background on why \(\displaystyle L^p\) spaces are a thing? What types of problems were they developed to solve? Where they just a sort of generalization of the Euclidean norm? The Analysis class I'm taking right now has very little context to it so I feel like I am learning random Mathematical facts.

 

[Ackbach]

$L^p$ spaces are a thing, I would say mostly, because of Quantum Mechanics. The only Hilbert space among the $L^p$ spaces is $L^2$, but that one Hilbert space is astronomically important. You essentially do quantum mechanics by analyzing operators, in particular doing spectral theory for the operator, on that Hilbert space. Or at least, that's one highly fruitful approach. $L^1$ is also important to QM, even though it's not a Hilbert space. So also is $L^{\infty}$.

These spaces have also been used in engineering, finance, and statistics, and probably a few other areas (pun intended, as usual). One of my former bosses once said that "All applied mathematics is basically operator theory on Hilbert space."

 

[Opalg]

As Ackbach has pointed out, the spaces $L^1$, $L^2$ and $L^\infty$ are intrinsically important, with applications all over the place. Other individual $L^p$-spaces occur in some applications. For example, if $p$ is an integer then the $L^p$-norm gives the $p$th moment of a random variable in probability theory. In PDE theory, $L^p$ conditions are useful in dealing with well-posedness and regularity issues. A non-integer value of $p$ occurs in Littlewood's 4/3 inequality, where the space $L^{4/3}$ makes an appearance.

More generally, the $L^p$-spaces form a natural "continuum" as $p$ goes from $1$ to $\infty$. The techniques of interpolation theory can be used to show that properties satisfied in $L^p$-spaces for some values of $p$ also hold for other values of $p$. A typical use of this technique would be to take something that is true in $L^1$ and $L^\infty$, and use interpolation to show that it also holds in $L^2$.