Abstract linear spaces
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It is possible however to trace the beginning of the vector concept back to the beginning of the 19th Century with the work of Bolzano. In 1804 he published a work on the foundations of elementary geometry Betrachtungen über einige Gegenstände der Elementargoemetrie
The move away from coordinate geometry was mainly due to the work of Poncelet and Chasles who were the founders of synthetic geometry. The parallel development in analysis was to move from spaces of concrete objects such as sequence spaces towards abstract linear spaces. Instead of substitutions defined by matrices, abstract linear operators must be defined on abstract linear spaces.
In 1827 Möbius published Der barycentrische Calcul
In 1837 Möbius published a book on statics in which he clearly states the idea of resolving a vector quantity along two specified axes.
Between these two works of Möbius, a geometrical work by Bellavitis was published in 1832 which also contains vector type quantities. His basic objects are line segments AB and he considers AB and BA as two distinct objects. He defines two line segments as 'equipollent' if they are equal and parallel, so, in modern notation, two line segments are equipollent if they represent the same vector. Bellavitis then defines the 'equipollent sum of line segments' and obtains an 'equipollent calculus' which is essentially a vector space.
In 1814 Argand had represented the complex numbers as points on the plane, that is as ordered pairs of real numbers. Hamilton represented the complex numbers as a two dimensional vector space over the reals although of course he did not use these general abstract terms. He presented these results in a paper to the Irish Academy in 1833. He spent the next 10 years of his life trying to define a multiplication on the 3-dimensional vector space over the reals. Hamilton's quaternions, published in 1843, was an important example of a 4-dimensional vector space but, particularly with Tait's work on quaternions published in 1873, there was to be some competition between vector and quaternion methods.
You can see a picture of the plaque commemorating where Hamilton discovered the Quaternions at THIS LINK and a (fanciful) engraving of when he carved the rules at THIS LINK.
In 1857 Cayley introduced matrix algebras, helping the move towards more general abstract systems by adding to the different types of structural laws being studied. In 1858 Cayley noticed that the quaternions could be represented by matrices.
In 1867 Laguerre wrote a letter to Hermite Sur le calcul des systèmes linéaires
Laguerre's work on linear systems was followed up by a paper by Carvallo in 1891. In this paper he defines operators on vector functions and draws a clear distinction between operators and matrices.
To understand the difference between the notions of an operator and a matrix, it suffices to say that, if one changes the coordinate system, one obtains a different matrix to represent the same vector function, but the same operator.Another mathematician who was moving towards geometry without coordinates was Grassmann. His work is highly original but the notion of barycentric coordinates introduced by Möbius was his main motivation. Grassmann's contribution Die Ausdehnungslehre
Grassmann studied an algebra whose elements are not specified, so are abstract quantities. He considers systems of elements on which he defines a formal operation of addition, scalar multiplication and multiplication. He starts with undefined elements which he calls 'simple quantities' and generates more complex quantities using specified rules.
But ... I go further, since I call these not just quantities but simple quantities. There are other quantities which are themselves compounded quantities and whose characteristics are as distinct relative to each other as the characteristics of the different simple quantities are to each other. These quantities come about through addition of higher forms ...His work contains the familiar laws of vector spaces but, since he also has a multiplication defined, his structures satisfy the properties of what are today called algebras. The precise structures are now known as Grassmann algebras. The ideas of linearly independent and linearly dependent sets of elements are clearly contained in Grassmann's work as is the idea of dimension (although he does not use the term). The scalar product also appears in Grassmann's 1844 work.
Grassmann's 1862 version of Die Ausdehnungslehre
Cauchy and Saint-Venant have some claims to have invented similar systems to Grassmann. Saint-Venant's claim is a fair one since he published a work in 1845 in which he multiples line segments in an analogous way to Grassmann. In fact when Grassmann read Saint-Venant's paper he realised that Saint-Venant had not read his 1844 work and sent two copies of the relevant parts to Cauchy, asking him to pass one copy to Saint-Venant.
However, rather typically of Cauchy, in 1853 he published Sur les clefs algébrique
The first to see the importance of Grassmann's work was Hankel. In 1867 he wrote a paper Theorie der complexen Zahlensysteme
The first to give an axiomatic definition of a real linear space was Peano in a book published in Torino in 1888. He credits Leibniz, Möbius's 1827 work, Grassmann's 1844 work and Hamilton's work on quaternions as providing ideas which led him to his formal calculus.
Peano's 1888 book Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva
In Chapter IX of the book Peano gives axioms for a linear space.
It is hard to believe that Peano writes the following in 1888. It could almost come from a 1988 book! The first is for equality of elements
- (a = b) if and only if (b = a), if (a = b) and (b = c) then (a = c).
- The sum of two objects a and b is defined, i.e. an object is defined denoted by a + b, also belonging to the system, which satisfies:
If (a = b) then (a + c = b + c), a + b = b + a, a + (b + c) = (a + b) + c,
and the common value of the last equality is denoted by a + b + c.
- If a is an object of the system and m a positive integer, then we understand by ma the sum of m objects equal to a. It is easy to see that for objects a, b, ... of the system and positive integers m, n, ... one has
If (a = b) then (ma = mb), m(a+b) = ma+mb, (m+n)a = ma+na,
m(na) = mna, 1a = a.
We suppose that for any real number m the notation ma has a meaning such that the preceeding equations are valid.
Peano goes on to state the existence of a zero object 0 and says that 0a = 0, that a - b means a + (-b) and states it is easy to show that a - a = 0 and 0 + a = a.
Peano defines a linear system to be any system of objects satisfying his four conditions. He goes on to define dependent objects and independent objects. He then defines dimension.
He proves that finite dimensional spaces have a basis and gives examples of infinite dimensional linear spaces. Peano considers entire functions f(x) of a variable x, defines the sum of f1(x) and f2(x) and the product of f(x) by a real number m. He says:-
Definition:The number of the dimensions of a linear system is the maximal number of linearly independent objects in the system.
If one considers only functions of degree n, then these functions form a linear system with n + 1 dimensions, the entire functions of arbitrary degree form a linear system with infinitely many dimensions.Peano defines linear operators on a linear space, shows that by using coordinates one obtains a matrix. He defines the sum and product of linear operators.
In the 1890's Pincherle worked on a formal theory of linear operators on an infinite dimensional vector space. However Pincherle did not base his work on that of Peano, rather on the abstract operator theory of Leibniz and d'Alembert. Like so much work in this area it had very little immediate impact and axiomatic infinite dimensional vector spaces were not studied again until Banach and his associates took up the topic in the 1920's.
Although never attaining the level of abstraction which Peano had achieved, Hilbert and his student Schmidt looked at infinite dimensional spaces of functions in 1904. Schmidt introduced a move towards abstraction in 1908 introducing geometrical language into Hilbert space theory. The fully axiomatic approach appeared in Banach's 1920 doctoral dissertation.
List of References (13 books/articles)
Additional Material in MacTutor
- Quaternion plaque
- Hamilton engraving
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|School of Mathematics and Statistics
University of St Andrews, Scotland