## Linear Methods of Applied Mathematics Evans M. Harrell II and James V. Herod*

#### *(c) Copyright 2000 by Evans M. Harrell II and James V. Herod. All rights reserved.

It was Descartes and Fermat who first discussed the vector spaces R2 and R3 in much the way they are presented today, but the emphasis was on points and graphing rather than on the concept of a vector. The notion of a vector is traceable to Bolzano, in its concrete form. The realization that abstract vector spaces abound in mathematics did not appear until the late nineteenth century. The modern definition seems to be due to the Italian mathematician Peano, who presented the modern form of the axioms of a vector space.

The focus on the important examples of function spaces as vector spaces is to be found in the work of Lebesgue and was formalized by Hilbert and Banach in the twentieth century.

Hilbert and Banach spaces are now core parts of graduate study in mathematics. Hilbert space refers to any inner-product space with the property that sequences with the Cauchy property have limits within the Hilbert space. This is necessary for doing analysis, and the main example is the function space L2 which plays a central rôle in this text.

More detail about the history of the notion of a vector space can be found at the MacTutor History of Mathematics site.