Interchange of Limiting Operations - Formulation

Formulation

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. This approach justifies, for example, the notion of uniform convergence. 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 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.