Linear Analysis(D)

LINEAR ANALYSIS (D)
Part IB Linear Algebra, Analysis II and Metric and Topological Spaces are essential.

Normed and Banach spaces. Linear mappings, continuity, boundedness, and norms. Finite-dimensional normed spaces. [4]
The Baire category theorem. The principle of uniform boundedness, the closed graph theorem and the inversion theorem; other applications. [5]
The normality of compact Hausdorﬀ spaces. Urysohn’s lemma and Tiezte’s extension theorem. Spaces of continuous functions. The Stone–Weierstrass theorem and applications. Equicontinuity: the Ascoli– Arzel`a theorem. [5]
Inner product spaces and Hilbert spaces; examples and elementary properties. Orthonormal systems, and the orthogonalization process. Bessel’s inequality, the Parseval equation, and the Riesz–Fischer theorem. Duality; the self duality of Hilbert space. [5]
Bounded linear operations, invariant subspaces, eigenvectors; the spectrum and resolvent set. Compact operators on Hilbert space; discreteness of spectrum. Spectral theorem for compact Hermitian operators. [5]

Appropriate books

† B. Bollob´as Linear Analysis. 2nd Edition, Cambridge University Press 1999
C. Goﬀman and G. Pedrick A First Course in Functional Analysis. 2nd Edition, Oxford University Press 1999
W. Rudin Real and Complex Analysis. McGraw–Hill International Editions: Mathematics Series

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