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?



Origin and first uses of $\ell_p$ norms?
Toeplitz in his [review](http://www.emis.de/cgi-bin/JFM-item?44.0401.01) of Riesz [[1913](http://archive.org/stream/lessystmesdq00riesuoft)] laments the lack of explicit motivation in generalizing from $\ell\_2$ to $\ell\_p$:



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> 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***.
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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](http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002501031)]: 



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> *If $\sum a\_kx\_k$ converges for all $x\in\ell\_p$, then $a\in\ell\_q$* (*and* $|\sum a\_kx\_k|\leqslant\|a\|\_q\|x\|\_p$).$(\*)$
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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](http://gallica.bnf.fr/ark:/12148/bpt6k31089/f35.image)] (resp. [[1912b](http://gallica.bnf.fr/ark:/12148/bpt6k31089/f498.image)]) for $q\in 2\mathbf Z$ and later by Hausdorff in general:



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> *If $\frac1p+\frac1q=1$ and $1<p\leqslant 2$, then* $\|c\|\_q\leqslant\|f\|\_p$ (*resp.* $\|f\|\_q\leqslant\|c\|\_p$). 
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**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](http://archive.org/stream/diophantischeapp00minkuoft#page/95)], but there is no such thing there. According to [this paper](http://pure.ltu.se/portal/en/publications/why-hoelders-inequality-should-be-called-rogers-inequality%2800c8b880-a576-11db-9811-000ea68e967b%29.html) the correct reference is Minkowski [[1896, p. 116](http://archive.org/stream/geometriederzahl00minkrich#page/n133)].
* [*Encyklopädie der math. Wiss.*, Vol. II.3.2](http://gdz.sub.uni-goettingen.de/dms/load/toc/?PPN=PPN360506542) (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*](http://www.digizeitschriften.de/en/dms/img/?PPN=PPN375375147_0001&DMDID=dmdlog10) (1994).
* According to [this paper](http://www.ams.org/mathscinet-getitem?mr=2347705), 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..."? 


