泛函分析   (91年度 下學期)

Chapter 1. Introduction-Sources of Functional Analysis and Preliminaries 1. Iteration method for Cauchy problem of ODE. Metric space. Banach contraction principle. 2. Separation of variables. Fourier series. Spaces 2 and L2 3. Sturm-Liouville problem and integral equations. 4. Elementary distribution theory and Green's functions. Chapter 2. Hilbert Spaces 1. Orthogonal projections. Orthonormal basis. Bessel inequality. Fourier expansion. 2. Riesz representation theorem. 3. Spectral theory for positive operators and Sturm-Liouville problem. 4. Spectral theory for compact self-adjoint operators and integral equations of Fredholm type. 5. Spectral theory for self-adjoint operators. Chapter 3. Banach spaces 1. Normed vector spaces. Hahn-Banach theorem. 2. Uniform boundedness principle. Open mapping principle. Closed graph theorem. Closed operators. 3. Compact operators. Fredholm alternative theorem. 4. Spectral theorem for bounded linear operators. Chapter 4. Banach Algebra and their Elementary Spectral Theory Normed algebras, Invertible elements, Resolvent, Spectral Radius. Chapter 5. Frechet Space, Introduction to Theory of Distribution Definitions and Examples, Operations on Distributions, Fourier Transform, Applications to Partial Differential Equations.

1. H. L. Royden, Real Analysis (3rd ed.) 2. Peter Lax, Functional Analysis, 2002 Wiley-Interscience. 3. R.Courant & D. Hilbert, Methods of Mathematical Physics, Vol. I.

20﹪習題，40﹪期中考，40﹪期末考。

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