Syllabus  MA490/MA482
MA481 Measure Theory and Functional Analysis

The Lebesgue integral: the deficiencies of the Riemann integral, Lebesgue
measure, measurable functions, the Lebesgue integral

Convergence theorems, functions of bounded variation and absolutely continuous
functions, Vitali's Covering Theorem, integration and differentiation

General measure and integration theory: outer measures, measures, measurable
functions, modes of convergence

Functional analysis: normed vector spaces and inner product spaces, bounded
linear mappings, linear functionals and the dual space, the classical Banach
spaces and their duals, Hilbert spaces, orthogonal decomposition, orthonormal
bases, Fourier series
Texts

H.L. Royden, "Real Analysis" (Collier Macmillan)

E. Kreyszig, "Introductory Functional Analysis with Applications" (Wiley)
References

R.G.Bartle, "The Elements of Integration" (Wiley)

A. Brown & C. Pearcey, Introduction to Operator Theory, Vol I (Springer)

E. Hewitt & K. Stromberg, "Real and Abstract Analysis" (Springer)

M. Spiegel, "Real Variables" (Schaum Outline)

W. Rudin, "Functional Analysis" (McGraw Hill)

W. Rudin, "Real and Complex Analysis" (McGraw Hill)

M. Schechter, "Principles of Functional Analysis" (Academic Press)

N. Young, "An Introduction to Hilbert Spaces" (Cambridge)
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