Interchange of limiting operations
Encyclopedia
In mathematics
, the study of interchange of limiting operations is one of the major concerns of mathematical analysis
. in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series.
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
.
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
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the study of interchange of limiting operations is one of the major concerns of mathematical analysis
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
. in that two given limiting operations, say L and M, cannot be assumed to give the same result when applied in either order. One of the historical sources for this theory is the study of trigonometric series.
Formulation
In symbols, the assumption- LM = ML,
where the LHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
means that M is applied first, then L, and vice versa on the RHS
Sides of an equation
In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. Each is solely a name for a term as part of an expression; and they are in practice interchangeable, since equality is symmetric...
, is not a valid equation
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
between mathematical operators, under all circumstances and for all operands. An algebraist would say that the operations do not commute
Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
. 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
Rigour
Rigour or rigor has a number of meanings in relation to intellectual life and discourse. These are separate from public and political applications with their suggestion of laws enforced to the letter, or political absolutism...
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
Well-behaved
Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...
for a given context. G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
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
Heuristic
Heuristic refers to experience-based techniques for problem solving, learning, and discovery. Heuristic methods are used to speed up the process of finding a satisfactory solution, where an exhaustive search is impractical...
, was that of Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
.
Examples
Examples abound, one of the simplest being that for a double sequence- am,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
- am,n = 2m − 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
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
also fall into this category: the symmetry of partial derivatives
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...
, differentiation under the integral sign, and Fubini's theorem
Fubini's theorem
In mathematical analysis Fubini's theorem, named after Guido Fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. As a consequence it allows the order of integration to be changed in iterated integrals.-Theorem...
deal with the interchange of differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
and integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
operators. One of the major reasons why the Lebesgue integral
Lebesgue integration
In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...
is used is that theorems exist, such as the dominated convergence theorem
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...
, that give perspicuous reasons why integration and infinite summation can be interchanged.