A Course in Operator Theory (Anglais) Relié – 15 octobre 1999
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Of course the calculus operators of grad/ div/ rad/ curl/ differentiate/ integrate and their twin cousins in analysis using functionals as well as functions extend beyond the current algebra research focus to geometry, Laplacians, Fourier and many other transforms, probability theory, and nearly every other area of math.
When metaphysics wonders if "math" is a tool used to understand the universe, or a tool used BY the universe, the serious side of the issue almost always involves the beauty, function and ubiquitous appearance of operators, which in lie algebras and quantum physics seem to somehow be part of the "fabric" of the paths themselves, not just their explanations or calculations!
This is a wonderful overview, but certainly not a "lay" or beginner treatment, and since it is very up to date even a decade later, the majority of the coverage IS algebras-- because that's where the majority of the "action" is even today in research. Of course quantum field theory also is actively researching other aspects of operators, particular in gauge analysis and theories. The treatment is conclusively graduate unless you're a very advanced undergrad who for some reason was really attracted to the calculus of variations and functional analysis, or algebraic mappings in general.
It's been said many times now that no one can really grasp the 600 or so current fields of math and how they relate-- and the price of this specialization is death according to Mr. Darwin. If there is an exception it's operators, and if there is any place to get a feel for the whole enchilada-- it's here. Stimulating not only as an intro to the field for folks who have mastered analysis and linear algebras and perhaps studied other algebras, geometries, etc., but also for those wanting to take a 30,000 foot view of all of math-- and one of the most ubiquitous "forces" in math, regardless of coordinate system, dimension, application, etc. If one other term crosses all those boundaries today, it is not surprising that it's the term "mapping" -- which goes hand in hand with operators.
Highly recommended both for courses/ course assistance and self study if you've got the above prereqs. Money saving tip: if you need to brush up on functional analysis or variations in general first, Dover has numerous older but still relevant titles to help us do so. A couple excellent ones are both Fomin translations: Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics) and Calculus of Variations (Dover Books on Mathematics).
If you are not familiar with John Conway-- as an author and teacher he is unequalled. His pedagogical style is the farthest thing from the "show off" style you'll encounter-- he takes the time, and makes the effort, to be sure you understand. From that view alone, this text is a delight.
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