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Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.
Rigor vs Intuition in Real Analysis
Real analysis is a branch of mathematical analysis which is concerned with the rigorous analysis of the real number system. Mathematical Analysis was the first area of mathematics in which the need and utility of the modern insistence on rigor was developed. Most mathematical concepts have been born out of intuitive ideas (think of how integration was developed through the idea of summing the areas an infinite amount of rectangles drawn under a curve). However as mathematics becomes more advanced this intuitive approach becomes limited and more difficult to verify.
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Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers, often including positive and negative infinity to form the extended real line.
Order properties of the real numbers
The real numbers have several important lattice-theoretic properties that are absent in the complex numbers. Most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity. Moreover, the ordering of the real numbers is total, and the real numbers have the least upper bound property. These order-theoretic properties lead to a number of important results in real analysis, such as the monotone convergence theorem, the intermediate value theorem and the mean value theorem.
However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in functional analysis and operator theory generalize properties of the real numbers – such generalizations include the theories of Riesz spaces and positive operators. Also, mathematicians consider real and imaginary parts of complex sequences, or by pointwise evaluation of operator sequences.
Relation to complex analysis
Real analysis is closely related to complex analysis, which studies broadly the same properties of complex numbers. In complex analysis, it is natural to define differentiation via holomorphic functions, which have a number of useful properties, such as repeated differentiability, expressability as power series, and satisfying the Cauchy integral formula.
In real analysis, it is usually more natural to consider differentiable, smooth, or harmonic functions, which are more widely applicable, but may lack some more powerful properties of holomorphic functions. However, results such as the fundamental theorem of algebra are simpler when expressed in terms of complex numbers.
Techniques from the theory of analytic functions of a complex variable are often used in real analysis – such as evaluation of real integrals by residue calculus.
Key concepts
Further information: Construction of the real numbersThe foundation of real analysis is the construction of the real numbers from the rational numbers. This is usually carried out by Dedekind–MacNeille completion, Dedekind cuts, or by completion of Cauchy sequences. Key concepts in real analysis are filters, nets, real sequences and their limits, convergence, continuity, differentiation, and integration. Real analysis is also used as a starting point for other areas of analysis, such as complex analysis, functional analysis, and harmonic analysis, as well as for motivating the development of topology, and as a tool in other areas, such as applied mathematics.
Important results include the Bolzano–Weierstrass and Heine–Borel theorems, the intermediate value theorem and mean value theorem, the fundamental theorem of calculus, and the monotone convergence theorem.
Various ideas from real analysis can be generalized from real space to general metric spaces, as well as to measure spaces, Banach spaces, and Hilbert spaces.
See also
- List of real analysis topics
- Time-scale calculus – a unification of real analysis with calculus of finite differences
- Real multivariable function
- Real coordinate space
- Complex analysis
Bibliography
- Aliprantis, Charalambos D.; Burkinshaw, Owen (1998). Principles of real analysis (3rd ed.). Academic. ISBN 0-12-050257-7.
- Browder, Andrew (1996). Mathematical Analysis: An Introduction. Undergraduate Texts in Mathematics. New York: Springer-Verlag. ISBN 0-387-94614-4.
- Bartle, Robert G.; Sherbert, Donald R. (2000). Introduction to Real Analysis (3rd ed.). New York: John Wiley and Sons. ISBN 0-471-32148-6.
- Abbott, Stephen (2001). Understanding Analysis. Undergradutate Texts in Mathematics. New York: Springer-Verlag. ISBN 0-387-95060-5.
- Rudin, Walter. Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw–Hill. ISBN 978-0-07-054235-8.
- Dangello, Frank; Seyfried, Michael (1999). Introductory Real Analysis. Brooks Cole. ISBN 978-0-395-95933-6.
- Bressoud, David (2007). A Radical Approach to Real Analysis. MAA. ISBN 0-88385-747-2.
- Kolmogorov, A. N.; Fomin, S. V. (1975). Introductory Real Analysis. Translated by Richard A. Silverman. Dover Publications. ISBN 0486612260. Retrieved 2 April 2013.
External links
- Analysis WebNotes by John Lindsay Orr
- Interactive Real Analysis by Bert G. Wachsmuth
- A First Analysis Course by John O'Connor
- Mathematical Analysis I by Elias Zakon
- Mathematical Analysis II by Elias Zakon
- Trench, William F. (2003). Introduction to Real Analysis (PDF). Prentice Hall. ISBN 978-0-13-045786-8.
- Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis
- Basic Analysis: Introduction to Real Analysis by Jiri Lebl
- Topics in Real and Functional Analysis by Gerald Teschl, University of Vienna.