This course consists of two (at times interrelated) parts:
Part I: Measure and integration
Part II: Linear operators, mostly on Hilbert spaces
Literature
- Tom L. Lindstr?m, Spaces — An introduction to Real Analysis, Pure and Applied Undergraduate Texts, vol. 29, American Mathematical Society, Providence, RI, 2017. Errata can be found here.
- Erik Bédos, Notes on Elementary Linear Analysis. The notes can be downloaded here.
The curriculum will consist of Chapter 7 and Chapter 8 (except sections 8.6–8.8) in Spaces plus the notes on linear analysis.
Alternative literature
It can often be helpful to look at other presentations of the curriculum. For the first part of the course, I recommend the following texts.
- Terence Tao, An introduction to measure theory, Graduate Studies in Mathematics, vol. 126, American Mathematical Society, Providence, RI, 2011.
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John N. McDonald and Neil A. Weiss, A course in real analysis, Academic Press, Inc., San Diego, CA, 1999.
Note that the material is organized differently than in Spaces, so a side-by-side comparison might prove difficult. My advice is to at least take a look at Section 2.1 on Problem solving techniques in Tao.
For the second part of the course, the following texts may be helpful and/or interesting.
- Barbara D. MacCluer, Elementary functional analysis, Graduate Texts in Mathematics, vol. 253, Springer, New York, 2009.
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Nicholas Young, An introduction to Hilbert space, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1988.