Fréchet Introduces the Concept of Abstract Space Summary

  • Last updated on November 10, 2022

Maurice Fréchet allowed points in abstract sets to be lines or curves, rather than geometric points, and defined a notion of distance between such objects, thereby generalizing the notion of distance in Euclidean three-dimensional space.

Summary of Event

By the late nineteenth century, abstraction and generalization had become significant trends in mathematics research. In his 1906 doctoral thesis, Maurice Fréchet considered abstract sets containing elements of a “general nature.” Prior to this, sets were regarded as collections of specific objects, such as numbers or points in the plane or collections of curves. In addition, Fréchet introduced a generalized notion of distance on his sets, which allowed him to use ideas from geometry in new ways. The combination of an abstract set and a distance on it (that is, an abstract metric space) was the starting point for two major areas of twentieth century mathematics: functional analysis and topology. Mathematics;abstract space Abstract space [kw]Fréchet Introduces the Concept of Abstract Space (1906) [kw]Abstract Space, Fréchet Introduces the Concept of (1906) [kw]Space, Fréchet Introduces the Concept of Abstract (1906) Mathematics;abstract space Abstract space [g]France;1906: Fréchet Introduces the Concept of Abstract Space[01490] [c]Mathematics;1906: Fréchet Introduces the Concept of Abstract Space[01490] Fréchet, Maurice Riesz, Frigyes Hadamard, Jacques-Salomon Volterra, Vito

The term “space” as used in mathematics usually refers to a set’s having some geometric structure. (Fréchet continued to use the term “abstract set,” ensemble abstraite, rather than “abstract space,” espace abstraite, until the mid-1920’s.) One can understand abstract space most easily by considering a sequence of problems. A single algebraic equation in one unknown usually has a solution consisting of a single number. When solving two equations in two unknowns, a solution may consist of a pair of numbers, which is considered as a point in the plane. Similarly, with three equations in three unknowns, a solution will often be three numbers and can be considered as a point in Euclidean three-dimensional space. If there are n equations in n unknowns, it is reasonable to look for n numbers as a solution and call the result a point in n-dimensional space. By the time of Fréchet, a generalization of this problem known as an integral equation was receiving attention. A solution to an integral equation is a function. (A function is a rule that takes input from one set and produces output in a possibly different set. A common example is the function that takes a number and produces the square of the number.) The set of functions that are solutions to integral equations can be thought of as an infinite dimensional space.

Fréchet’s ultimate goal was to study functionals—that is, functions defined on spaces of functions. (An example is the functional that takes a curve and produces its length.) His novel approach was to ignore initially the individual aspects of the classes of functions and work instead on abstract sets. His first task was to impose some geometric structure on an abstract set. Geometry is involved because functions can be represented often by lines or curves in the plane or as surfaces in three dimensions. Although the idea of giving a class of functions a “geometrical” structure can be traced back to George Bernhard Riemann’s doctoral thesis in 1851, Fréchet’s contribution was to use ideas from ordinary three-dimensional geometry as motivation to ask questions about spaces of larger (and even infinite) dimension.

As with most new ideas in mathematics, Fréchet’s ideas were motivated by the problems of the day. One such source was the work of Vito Volterra in the calculus of variations. An early problem in this area was to find the curve in the plane having a fixed length that encloses the largest area. The answer was a circle, but the rigorous verification of this (and the solution of related problems) was far from obvious. It was necessary to take an arbitrary closed curve (with a fixed perimeter) and compute the enclosed area. This process described a function: For each curve, produce its area. This is an example of what Volterra called a “function of lines” (now called a functional) and is a typical example of a functional acting on an abstract set.

Fréchet wrote his thesis, “Sur Quelques Points du calcul fonctionnel” (1906; on some points of functional calculation), in Paris at the École Normale Supérieure under the direction of Jacques-Salomon Hadamard, who was known for his work in partial differential equations. The first of two parts consisted of the preliminary steps toward what is now called point set topology. The goal was to be able to define the limit of a sequence of elements in an abstract set. Fréchet did this in three ways of varying generality. His third approach—and the only one that has flourished—was through what Fréchet called an écart. This construction is now called a metric, so named by Felix Hausdorff in his famous book on set theory and topology, Grundzüge der Mengenlehre (1914; fundamentals of set theory).

For each pair of points x and y in a set, a metric Mathematics;metrics assigns a nonnegative number to represent the distance between the points. This number (that is, distance) is denoted by d(x, y). Fréchet’s notation is simply (x, y). It satisfies the following simple properties: The distance from x to y is the same as that from y to x; d(x, y) is equal to zero if and only if x equals y; and the triangle inequality—the distance from x to y is less than (or equal to) the sum of the distances from x and y to another point z. Using this idea of distance, Fréchet was able to define continuity in abstract sets, which enabled him to apply his new ideas to some classical spaces of functions. (A function is continuous if input points that are “close together” produce output that is “close.” Geometrical properties of the input are transferred to the output by continuous functions.)

The topological concepts of compactness, completeness, and separability were important elements of Fréchet’s thesis. The first of these is especially important for solutions of equations in infinitely many unknowns. If a sequence of elements in a set is converging, the limiting element is guaranteed to exist in the set if the set is known to be compact. This allows one to solve equations by producing a sequence of approximate solutions; the actual solution is then the limit of the approximations. This idea was used long before Fréchet, and several conditions were known to guarantee the existence of the limit solution. The works of Cesare Arzelà and Giulio Ascoli on equicontinuity had great influence on Fréchet’s thesis in this regard, but it was Fréchet who was able to isolate the essential ideas and unite them in an abstract framework. Fréchet also applied the abstract concepts of compactness, completeness, and separability to concrete examples of spaces of functions.

Another important mathematician who was influential in creating the notion of abstract space was Frigyes Riesz. His 1907 paper on the origins of the concepts of space was in large measure a quasi-philosophical paper on a mathematical model for the geometry of space as needed in physics. Riesz, however, introduced the concepts of derived set, neighborhood, and connectedness, which came to be standard ideas in Hausdorff’s topology. Fréchet’s work was an important stimulus for future work of Riesz in his development of the notion of the function space consisting of square integrable functions and a metric on this space of functions. Both Fréchet and Riesz published theorems providing concrete representations of certain kinds of abstract linear functionals. This early work of Riesz and Fréchet, motivated by Hadamard, was the beginning of a fruitful area of modern mathematics known as functional analysis.

Significance

The immediate effect of Fréchet’s thesis was to initiate the study of point set topology and provide tools for early work in functional analysis. Fréchet’s definition of metric space is the one used today, and the study of metric spaces is still ongoing. Not all of the concepts that Fréchet defined were sufficiently powerful to solve the relevant problems of the day; they needed further refinement by others. For example, his notion of compactness was not powerful enough to treat convergence problems outside of metric spaces, and other mathematicians (for example, Hausdorff) needed to generalize the concept. It was also found that basing the concept of convergence on sequences was too restrictive, and the idea of neighborhood proved more fruitful. (Fréchet, Riesz, Hausdorff, and others made advances in this direction between 1906 and 1920.) Rather than being remembered for his theorems and their applications, Fréchet is instead best known for the objects he defined and his approach to abstract space. Although he continued to work in the field of topology and analysis until about 1930 (at which time he moved into the study of probability), Fréchet never produced another work that was more influential than his thesis.

The abstract, axiomatic approach to mathematics that is evident in Fréchet’s thesis continued and reinforced a growing tendency in mathematics. “Axiomatics” refers to the development of mathematics through precise statements of axioms and definitions followed by the logical consequences of the axioms—careful statements and proofs of theorems. Classical Euclidean geometry has been presented axiomatically for centuries, but the rest of mathematics did not begin to develop this way until the late nineteenth century. Beginning in earnest with the work of David Hilbert at the beginning of the twentieth century, mathematics developed axiomatically.

The axiomatic aspect of Fréchet’s work was especially important for the future of functional analysis. This area of mathematics, originating in the early 1900’s, is characterized by the application of the abstract approaches of algebra and topology to problems in classical mathematics, such as differential and integral equations, as well as to parts of physics, such as quantum mechanics. It is important for the development of functional analysis to have an abstract view of the problems so as to extract their essential and common elements. It is precisely this point of view that Fréchet helped popularize. Mathematics;abstract space Abstract space

Further Reading
  • citation-type="booksimple"

    xlink:type="simple">Dieudonné, Jean. History of Functional Analysis. New York: North-Holland, 1981. The author is one of the important mathematicians of the twentieth century. Several of the later chapters in this book contain technical material, but the other chapters are for the most part accessible to nontechnical readers.
  • citation-type="booksimple"

    xlink:type="simple">Fréchet, Maurice. “From Three-Dimensional Space to the Abstract Spaces.” In Great Currents of Mathematical Thought, edited by François le Lionnais. New York: Dover, 1971. One of the few papers written by Fréchet available in English. Provides basic background and motivation for considering abstract space. Also includes a discussion of metric spaces, topological vector spaces, and applications to functional analysis.
  • citation-type="booksimple"

    xlink:type="simple">Kline, Morris. Mathematical Thought from Ancient to Modern Times. Vol. 3. 1972. Reprint. New York: Oxford University Press, 1990. This third volume of a monumental, well-written work contains two chapters relevant to Fréchet’s work. Chapter 46 gives background and an outline of functional analysis, and chapter 50 treats the beginnings of topology. Suitable for both the general reader and the professional mathematician.
  • citation-type="booksimple"

    xlink:type="simple">Taylor, Angus E. “A Study of Maurice Fréchet: I. His Early Work on Point Set Theory and the Theory of Functionals.” Archive for the History of Exact Sciences 27 (1982): 233-295. Contains a wealth of information about Fréchet’s life and work as well as about other mathematicians of his time.
  • citation-type="booksimple"

    xlink:type="simple">Temple, George. One Hundred Years of Mathematics: A Personal Viewpoint. New York: Springer-Verlag, 1981. The first few sections of chapter 9 discuss the origins of topology, and some later sections of the chapter treat the foundations of functional analysis. More technical than biographical, but still accessible to the interested nonspecialist.

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