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Advanced Real Analysis deals with topics in real analysis that the young
mathematician needs to know in order to communicate well with colleagues
in all branches of mathematics. These topics include parts of Fourier analysis,
functional analysis, spectral theory, distribution theory, abstract harmonic analysis,
and partial differential equations. They tend to be ones whose applications
and ramifications cut across several branches in mathematics. Each topic can
be studied on its own, but the importance of the topic arises from its influence
on the other topics and on other branches of mathematics. To avoid having all
these relationships come across as a hopeless tangle, the book needed some
organizational principle for its design. The principle chosen was largely to
organize topics according to their role in the study of differential equations. This
organizational principle influences what appears below, but it is certainly not
intended to suggest that applications to differential equations are the only reason
for studying certain topics in real analysis.
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