Interchange of limiting operations

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In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis. While analysis is characterised by the use of mathematical limits, in many guises, there is an inherent difficulty in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order.

In symbols, the assumption

LM = ML,

where the LHS means that M is applied first, then L, and vice versa on the RHS, is not a valid equation between mathematical operators, under all circumstances and for all operands. An algebraist would say that the operations do not commute. The approach taken in analysis is somewhat different. Conclusions that assume limiting operations do 'commute' are called formal. The analyst tries to delineate conditions under which such conclusions are valid; in other words mathematical rigour is established by the specification of some set of sufficient conditions for the formal analysis to hold good. It is relatively rare for such sufficient conditions to be also necessary, so that a sharper piece of analysis may extend the domain of validity of formal results.

Professionally speaking, therefore, analysts push the envelope of techniques, and expand the meaning of well-behaved for a given context. G. H. Hardy, in an Appendix A note on double limit operations to A Course of Pure Mathematics, wrote that "The problem of deciding whether two given limit operations are commutative is one of the most important in mathematics". An opinion apparently not in favour of the piece-wise approach, but of leaving analysis at the level of heuristic, was that of Richard Courant.

Examples abound, one of the simplest being that for a double sequence

a_m,n

it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. For example take

a_m,n = 2^{m − n}

in which taking the limit first with respect to n gives 0, and with respect to m gives ∞. Replacing all values greater than 1 by 1, the limits can be made to be 0 or 1 according to the order.

Many of the fundamental results of infinitesimal calculus also fall into this category: the symmetry of partial derivatives, differentiation under the integral sign, and Fubini's theorem deal with the interchange of differentiation and integration operators. One of the major reasons why the Lebesgue integral is used is that theorems exist, such as the dominated convergence theorem, that give perspicuous reasons why integration and infinite summation can be interchanged.