ELEMENTARY REAL ANALYSIS, thomson bruckner
For a course at this level, though, “real analysis” mostly refers to the
subject matter that you have already learned in your calculus courses: limits, continuity, derivatives, integrals, sequences, and series.Calculus as a
subject can be thought of as an eighteenth century development, analysis
as a nineteenth-century creation.None of the ideas of calculus rested on
very firm foundation, and the lack of foundations proved a barrier to further progress.There was much criticism by mathematicians and philosophers of
the fundamental ideas of calculus (limits especially), and often when new
and controversial methods were proposed (such as Fourier series) the mathematicians of the time could not agree on whether they were valid.
The topics of the books are given below:
1. PROPERTIES OF THE REAL NUMBERS
2. SEQUENCES
3. INFINITE SUMS
4. SETS OF REAL NUMBERS
5. CONTINUOUS FUNCTIONS
6. MORE ON CONTINUOUS FUNCTIONS AND SETS
7. DIFFERENTIATION
8. THE INTEGRAL
9. SEQUENCES AND SERIES OF FUNCTIONS
10. POWER SERIES
11. THE EUCLIDEAN SPACES
12. DIFFERENTIATION ON EUCLIDEAN SPACES
13. METRIC SPACES
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