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 R^{2} and
R^{3} 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 L^{2}
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.

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Evans M. Harrell II (correct my scholarship!)