Origin and first uses of $\ell_p$ norms? 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?
 A: Dieudonné and Pietsch both believe that Frederic Riesz was the one to introduce the spaces $l^p$ in his monograph
"Les systèmes d'équations linéaires à une infinité d'inconnues", 1913.
This seems to be available online, which is good because unfortunately I cannot read French very well... 
It seems that indeed, he develops here the theory of $l^p$-spaces, much later than the spaces $L^p$, which were considered by him in
"Untersuchungen über Systeme integrierbarer Funktionen" in 1909.
In the first two chapters of his monograph, Riesz mentions motivations for his work: infinite systems of equations, infinite determinants, theorems of Landau and Pringsheim. Someone who can read better French could expand my list...
A: Toeplitz in his review of Riesz [1913] laments the lack of explicit motivation in generalizing from $\ell_2$ to $\ell_p$:

The considerations of the 3rd Chapter require not the convergence of the sum of squares of the unknowns, but the more general convergence of $\sum|x_a|^p$, where $1<p<\infty$ (even the limiting cases $p = 1$, $p = \infty$ are discussed), a generalization on which the author seems to place considerable value, but whose deeper analytic significance he doesn't further motivate.

But isn't the motivation simply that this increases the chances for a system $\sum a_{ik}x_k=c_i$ to have a solution? Namely (if $p>2$ say) we are allowed to look for solutions in the wider space $\ell_p$ — but the price to pay is that we must know that each "row" $a_{i\,\cdot}$ is in $\ell_q$ where $\frac1p+\frac1q=1$. That seems to be the point of the theorem that Riesz [1913, p. 47] attributes to Landau [1907]: 

If $\sum a_kx_k$ converges for all $x\in\ell_p$, then $a\in\ell_q$ (and $|\sum a_kx_k|\leqslant\|a\|_q\|x\|_p$).$(*)$

This, in retrospect, is essentially the proof that $\ell_p$ has dual $\ell_q$, and I would say it qualifies as a use of $\ell_p$ norms predating Riesz. But as to who first used this to solve a concrete problem...? I don't know.
Another pre-Riesz $\ell_p$ result is the Hausdorff-Young inequalities for Fourier series $f(e^{i\theta})=\sum c_ne^{in\theta}$, proved by Young in [1912a] (resp. [1912b]) for $q\in 2\mathbf Z$ and later by Hausdorff in general:

If $\frac1p+\frac1q=1$ and $1<p\leqslant 2$, then $\|c\|_q\leqslant\|f\|_p$ (resp. $\|f\|_q\leqslant\|c\|_p$). 

More secondary sources — in addition to the Dieudonné and Pietsch texts cited by András: 


*

*Riesz [1913, p. 45] attributes the Minkowski inequality to Minkowski [1907, p. 95], but there is no such thing there. According to this paper the correct reference is Minkowski [1896, p. 116].

*Encyklopädie der math. Wiss., Vol. II.3.2 (1927) reports on both Riesz's $\ell_p$ theory with forerunners Hölder and Landau (p. 1445, no mention of Minkowski), and on Hausdorff-Young (pp. 1211-1212). 

*A more recent survey: The Hausdorff-Young theorems of Fourier analysis and their impact (1994).

*According to this paper, Hölder's inequality $(*)$ goes back to Grolous (1875) and Besso (1879).
Yet again, none of these references really address the motivation for generalizing from $\ell_2$ to $\ell_p$. Maybe it was a case of "because we can..."?  
