It seems G. H. Hardy once wrote that "The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics". I was wondering what led to the need to formulate the question of when the limit operations commute.

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    $\begingroup$ Not a historian of mathematics at all but an untuitively obvious case where double limits pop up is in iterated integrals (i.e. the ones that allow you to actually compute volume integrals). $\endgroup$
    – WoolierThanThou
    Mar 7, 2021 at 10:08
  • $\begingroup$ The quote is from my favorite A Course of Pure Mathematics by G H Hardy (see Appendix 3, A note on double limit problems). $\endgroup$ Mar 7, 2021 at 15:31
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    $\begingroup$ I think in time of Euler this was a common technique and people did not bother to check the conditions which ensure commutativity of two limit operations. But later as focus on rigor increased, such issues were analyzed in some detail. In particular a significant portion of development of theory of integration is based on exchanging limit with integral. $\endgroup$ Mar 7, 2021 at 15:36
  • $\begingroup$ This seems to be an immediate questioning that comes when you consider a "limit of a limit". $\endgroup$
    – Yves Daoust
    Mar 7, 2021 at 19:38
  • $\begingroup$ See Bressoud’s books “A radical approach to real analysis” and “A radical approach to Lebesgue’s theory of integration” for background about convergence issues that motivated the development of various topics in analysis. $\endgroup$
    – KCd
    Mar 10, 2021 at 21:36

4 Answers 4


G. H. Hardy wrote in A Course in Pure Mathematics in Appendix II, section A note on double limit problems

  • Let us consider some special instances. In $\S\ 213$ we proved that \begin{align*} \log(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots \end{align*} where $-1<x\leq 1$, by integrating the equation \begin{align*} 1/(1+t)=1-t+t^2-\cdots \end{align*} between the limits $0$ and $x$. What we proved amounted to this, that \begin{align*} \int_{0}^{x}\frac{dt}{1+t}=\int_{0}^{x}\,dt-\int_{0}^{x}t\,dt+\int_{0}^{x}t^2\,dt-\ldots \end{align*} or in other words that the integral of the sum of the infinite series $1-t+t^2-\cdots$ taken between the limits $0$ and $x$, is equal to the sum of the integrals of its terms taken between the same limits. Another way of expressing this fact is to say that the operations of summation from $0$ to $\infty$, and of integration from $0$ to $x$, are commutative when applied to the function $(-1)^nt^n$, i.e. that it does not matter in what order they are performed on the function.

he continues with another important example:

  • ... we proved that the differential coefficient of the exponential function \begin{align*} \exp x=1+x+\frac{x^2}{2!}+\cdots \end{align*} is itself equal to $\exp x$, or that \begin{align*} D_x\left(1+x+\frac{x^2}{2!}+\cdots\right)=D_x1+D_xx+D_x\frac{x^2}{2!}+\cdots; \end{align*} that is to say that the differential coefficient of the sum of the series is equal to the sum of the differential coefficients of its terms, or that the operations of summations from $0$ to $\infty$ and of differentiation with respect to $x$ are commutative when applied to $x^n/n!$.

The author accompanies these examples with some more and concludes then in convincing manner:

The really important case (as is suggested by the instances which we gave from Ch. IX) is that in which each operation is one which involves a passage to the limit, such as a differentiation or the summation of an infinite series: such operations are called limit operations. The general question as to the circumstances in which two given limit operations are commutative is one of the most important in all mathematics. ...

We can already defer from the examples above that exchanging limits is of fundamental importance and seemingly ubiquituous in analysis. So, the question for historical origin of the need of exchanging limits is rather hard to answer.

Maybe one could trace back some lines of history by looking at the history of named theorems which are based upon the exchange of limits, like Fubini's Theorem ,Theorem of Schwarz, etc.

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    $\begingroup$ @A.D.: Regarding some history behind two types of limit interchanges, see my comments to this answer (mainly, the two sci.math posts I cite). $\endgroup$ Mar 7, 2021 at 17:28
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    $\begingroup$ +1 for putting the relevant passages from Hardy's note here. $\endgroup$ Mar 7, 2021 at 18:50
  • $\begingroup$ @ParamanandSingh: Many thanks for the upvote. At first I was looking for some good examples in some of Courant's books, cause he provides typically very nice and enlightning explanations. But then I was lucky to find the relevant passages from G.H. Hardy itself, who is also a master in pedagogically valuable explanations. :-) $\endgroup$ Mar 7, 2021 at 18:57
  • $\begingroup$ @MarkusScheuer: The Wikipedia page for "Interchange of limiting operations" has in it the following sentence: "One of the historical sources for this theory is the study of trigonometric series." So it looks like, for some reason, one of the places people first noticed the fact that this was an issue to be made rigorous in that context? $\endgroup$
    – A.D.
    Mar 9, 2021 at 5:32

Once you're working with $\ge 2$ independent variables, it's necessary anytime you need to be able to commute differential/integral operators. With differentials, for example, we need it in order to change the order in which we take partial derivatives:

$$ \frac{\partial }{\partial x}\left( \frac{\partial f}{\partial y}\right) = \frac{\partial }{\partial y}\left( \frac{\partial f}{\partial x}\right) $$

With integrals, it's necessary anytime we want to take double (or larger) integrals: $$ \int_{X \times Y} f(x,y) \ d(x,y) = \int_X \left( \int_Y f(x,y) \ dy \right) dx = \int_Y \left( \int_X f(x,y) \ dx \right) dy $$

It also plays a role when working with limits of derivatives or limits of integrals, as for example in Hardy's own work.


Here is one example. The question of whether limits commute arises when we ask whether the limit of a sequence of continuous functions is continuous. If each $f_n$ is continuous, and $f_n \to f$ pointwise, then it’s not necessarily true that $f$ is continuous. But the result is true if the convergence is uniform. With uniform convergence, we can interchange limits.

From Walter Rudin’s Principles of Mathematical Analysis, 3rd edition, p.144:

      The main problem which arises is to determine whether important properties of functions are preserved under the limit operations (1) and (2). For instance, if the functions $f_n$ are continuous, or differentiable, or integrable, is the same true of the limit function? What are the relations between $f_n’$ and $f’$, say, or between the integrals of $f_n$ and that of $f$?

      To say that $f$ is continuous at $x$ means $$ \lim_{t \to x} f(t) = f(x). $$

Hence, to ask whether the limit of a sequence of continuous functions is continuous is the same as to ask whether $$ \lim_{t \to x} \lim_{n \to \infty} f_n(t) = \lim_{n \to \infty} \lim_{t \to x} f_n(t), $$ i.e., whether the order in which the limit processes are carried out is immaterial.

From the Wikipedia page on uniform convergence, there is a historical note:

In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous, to which Niels Henrik Abel in 1826 found purported counterexamples in the context of Fourier series, arguing that Cauchy's proof had to be incorrect. Completely standard notions of convergence did not exist at the time, and Cauchy handled convergence using infinitesimal methods. When put into the modern language, what Cauchy proved is that a uniformly convergent sequence of continuous functions has a continuous limit. The failure of a merely pointwise-convergent limit of continuous functions to converge to a continuous function illustrates the importance of distinguishing between different types of convergence when handling sequences of functions.


The finite case is the simplest and is easily seen by observation. If we set out an array of pebbles with differing column and rows sizes, we can see just by observation that the total number of pebbles must be the same whether we count by rows or by columns.

This would be appear to generalise to the infinite case where we take a pear and infinitely slice in the vertical and horizontal case. Then adding up the infinitesimal cubes first in the horizontal direction and then in the vertical should gives us back tge pear; lilewise when the two operations are swapped, that is we first add the infinitesinal cubes in the vertical and then the horizontal directions.

However, this is in fact no longer true in the infinite case, at least without qualification. This is where analysis comes in, and specifically, the Fubini-Tonelli theorem on swapping limits. Since Hardy was an analyst, it may have been this theorem he was thinking off.


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