Ronnie points out the very excellent 1997 book on smooth analysis by Kriegl & Michor. In fact, not the reviewer but the authors themselves originally stated the principle of functional analysis "rather than" category theory. It is rather strange since much of the material in the book was arrived at by very categorical means. For example, results published in Kriegl's joint work with Alfred Frolicher are basic. My dismay is reflected in my RCMP paper on Volterra, where I praise the book for its powerful combination of functional analysis "and" category theory. In a related expositional choice the book claims to be about topological vector spaces, but the definition of morphism used betrays the fact that the weaker structures of bounded sequences and of C-infinity paths are the actual underpinning. It would be instructive to know whether this strategy actually widened the audience in the past 10 years. Bill On Thu Mar 6 10:15 , "Ronnie" sent:
You could also look at MR1471480 (98i:58015) Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis. (English summary) Mathematical Surveys and Monographs, 53. American Mathematical Society, Providence, RI, 1997. x+618 pp. ISBN: 0-8218-0780-3
for which an e-version has been downloadable. However as the review says: "the exposition is based on functional analysis rather than on category theory; this fact will, undoubtedly, allow the subject to reach a wider audience. "
Ronnie
----- Original Message ----- From: "Robert L Knighten" RLK@knighten.org> To: "Categories list" categories@mta.ca> Sent: Thursday, March 06, 2008 2:37 AM Subject: categories: How to motivate a student of functional analysis
Most of the material connecting analysis and category theory seems to be written by specialists in category theory who have observed some of the ways that insights from category theory can be brought to bear (for example look on Math Sci Net at the many papers (co)authored by Joan Wick Pelletier.) But here's an example which is a book on functional analysis that has a strong use of categories:
@book {MR0296671, AUTHOR = {Semadeni, Zbigniew}, TITLE = {Banach spaces of continuous functions. {V}ol. {I}}, NOTE = {Monografie Matematyczne, Tom 55}, PUBLISHER = {PWN---Polish Scientific Publishers}, ADDRESS = {Warsaw}, YEAR = {1971}, PAGES = {584 pp. (errata insert)}, MRCLASS = {46E15 (46M99)}, MRNUMBER = {MR0296671 (45 \#5730)}, MRREVIEWER = {H. E. Lacey}, }
-- Bob
-- Robert L. Knighten RLK@knighten.org
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