When exactly were $\ell_p$ norms first defined and used?

(Here is what I know, or think I know: Lebesgue and/or Riesz had something to do with them, but in some sense they go back to Minkowski, since Minkowski's inequality is (in essence) the statement that an $\ell_p$ norm is a norm.)

Here is what is really my main question: how were $\ell_p$ norms ($p\geq 1$ arbitrary) first used? What was their motivation? It is clear that $\ell_1$, $\ell_2$ and $\ell_\infty$ norms are very natural, and their use long predates the formal definition of "form". The $\ell_4$ norm also pops up on its own sometimes. In contrast, $\ell_p$ norms for other $p$ seem to arise most often in the course of a proof, as a tool, when one needs some notion of "size" that falls between an $\ell_1$ and an $\ell_2$ norm (for example). Did the first uses of $\ell_p$ norms fit this framework? Can you think of some interesting (and preferably early) instances that do not obey this pattern?

History of Banach spaces and linear operatorsmay be of interest. The references cited in Subsection 4.4.6 seem to be amongst the oldest cited in that section of Pietsch's book. $\endgroup$"Note that the concept of a norm was not yet in use, though Riesz proved Minkowski's inequality."$\endgroup$2more comments