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