A general outline of the genesis of vector space theory
The paper is what the title says, an outline; it is not an everywhere thorough investigation. Though vector space theory is quite common today, its history is not yet well known. This is the reason, why the list of references is mainly based on sources, there really does not exist much secondary literature. From antiquity up to the end of the 19th century linear methods were adopted in different branches of mathematics but there was no unification, no focusing on the main points, there was no structure like definition. One of these branches was the theory of linear equations. Though in 1750 Euler and Cramer made first steps in direction of a theory of determinants, they neglected undetermined and inconsistent systems of linear equations. Only between 1840-1879 the ``rank'' of a determinant was defined and became a central point in the description of linear equations, the final results are due to Frobenius. A second branch was the concept of vector in geometry. In this context there are mentioned Möbius and Bellavitis, Bellavitis being the first to define an addition of vectors in space. But, according to the author, the elaboration of geometric calculus seems to have had more influence on the development of vector analysis than on the theory of vector space (p. 237). Grassmann's Ausdehnungslehre is only regarded as an isolated singularity (p. 241-246). Further impulses are due to ``linear algebra in the dialectic exchanges between geometry and algebra'' (p. 237-241), these are mainly the contributions to the theory of transformations, i.e. the results of Euler, Lagrange, Gauss, Cayley and Sylvester. In the second half of the 19th century there existed a geometry of \(n\) dimensions, developed on the basis of analytic geometry and the theory of determinants and matrices. The first axiomatic approaches were made by Peano and his school and Weyl (1888, 1918). In 1893 Dedekind gave a precise definition of a linear structure and in 1910 Steinitz contributed the definition of linear dependence. In this regard also the theory of differential equations and functional analysis has to be mentioned. In 1822 Fourier solved differential equations in infinite unknowns and in 1886 Poincaré investigated infinite systems of linear equations. They were succeeded by Hilbert, Riesz, Schmidt, Hadamard and Fréchet. One of the most important papers was published by Riesz. He was the first to give a definition of a normal and of a closed vector subspace. Now time had come for textbooks presenting all these new results: van der Waerden (1930), Birkhoff and Mac Lane (1941), and Bourbaki (1947).
scientific article; zbMATH DE number 801052
scientific article