Mathematics around Takeo Nakasawa

Abstract

Nowadays every freshman or sophomore specialized in science, engineering and economics is required to study linear algebra as well as advanced calculus. Nevertheless we should not forget that linear algebra is a relatively new field in the very long history of mathematics. Linear algebra was once known as the theory of matrices and determinants. It took a long period for mathematicians to accept non-numerical entities as objects of their own research. Therefore it is not surprising to note that determinants were introduced long before matrices were exactly formulated, because determinants are numerical entities, though obtained through relatively complicated processes, but matrices are not. It was in the latter half of the 17th century that two great mathematicians discovered determinants independently, one in Japan and the other in Hanover, which lies in the Western part of Germany now. The former was called Takakazu Seki, and the other was called Gottfried Wilhelm Leibniz. Unfortunately their discoveries did not have a great influence on the main development of mathematics. What is now called the formula of Cramer was announced by Cramer in 1750. It was Cauchy that has established a systematic theory of determinants in 1812. In 1858 Cayley published a paper containing a result which is now called the Cayley-Hamilton theorem. What is now called the Jordan standard form of a square matrix was discovered by Jordan in 1871, when he was keenly aware of its applications in the theory of linear differential equations. It was Frobenius that has introduced the notion of linear independence in solutions for linear equations in 1879. The abstract notion of a vector space was introduced by Peano in 1888, though it attracted little attention at that time, and it was reintroduced by later mathematicians such as Steiniz in the context of algebraic extensions of fields in 1910. It is very interesting to note that it was only in the beginning of the 20th century that a textbook on linear algebra became available, so that linear algebra was presumably an exciting subject to Takeo Nakasawa in the 1930s. Stephan Banach introduced the notion of a Banach space in his doctoral thesis in 1920, and the term of a vector space has found itself in the vocabulary of every mathematician when Banach’s pioneering book on the theory of linear operators was published a decade later. Strange to say, the burgeoning theory of infinite-dimensional vector spaces made the notion of a vector space accepted by the general mathematical community. This is presumably because finite-dimensional linear algebra can be formulated as the theory of matrices and determinants without any reference to vector spaces at all, but the theory of linear operators on infinite-dimensional vector spaces is forced to begin with the very definition of an infinite-dimensional vector space.