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?

  • 2
    $\begingroup$ I don't really know the answer, but Section 4.4 of Pietsch's book History of Banach spaces and linear operators may 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. $\endgroup$ Jul 10, 2013 at 11:19
  • $\begingroup$ In terms of when they were first considered as 'norms', the following remark from p.3 of Pietsch's book may be noteworthy: "Note that the concept of a norm was not yet in use, though Riesz proved Minkowski's inequality." $\endgroup$ Jul 10, 2013 at 11:25
  • $\begingroup$ Explicit considerations of p-norm distances for $p=3$, $p=4$ or $p=5$ emerge quite naturally in the field of data clustering. Unfortunately, I don’t know any early paper on this but it seems that such issues had to be considered almost from the beginning of the field. This paper can be seen as a modern example of such considerations: gracehopper.org/2004/Proceedings/PDF/ni_Vadapalli.pdf $\endgroup$
    – Waldemar
    Jul 10, 2013 at 14:00
  • 1
    $\begingroup$ This also doesn't answer your question, but you may be interested to know there are results that characterize $\ell_p$ in natural ways. See golem.ph.utexas.edu/category/2011/03/… $\endgroup$ Jul 13, 2013 at 12:06
  • 1
    $\begingroup$ I don't really see why talk of the Plancherel formula has any relevance to HAH's question "how were $\ell^p$ norms ($p\geq 1$ arbitrary) first used?" - but then perhaps I am being overly literal $\endgroup$
    – Yemon Choi
    Jul 18, 2013 at 23:54

2 Answers 2


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:

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..."?

  • 1
    $\begingroup$ Hausdorff-Young and duality make general $p\ne 2$ of interest if you are already interested in some $p\ne 2$... $\endgroup$ Jul 19, 2013 at 18:06
  • 2
    $\begingroup$ Exactly. And so I feel that I have failed to find any early use of $\ell_p$ norms in the sense you seem to want: an application to something else. It might help if you gave (not necessarily early) examples of what you consider significant uses or applications: then one could try to trace back their history. $\endgroup$ Jul 19, 2013 at 18:20
  • $\begingroup$ I believe Riesz's reference to page 95 of Minkowski is a misprint for page 85 where Minkowski shows that convexity of a solid containing the origin is equivalent to homogeneity and the triangle inequality of its associated gauge functional (equations (37), (39)). The convexity of the two-dimensional $\ell_p$-balls is proved on page 48, where there is the familiar picture of the nested $\ell_p$ unit balls in the plane. $\endgroup$
    – Martin
    Jul 20, 2013 at 7:00

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...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.