I.Contents：
Chapter 1. Metric Spaces
1.Locally compact and compact Sets.
2.Baire Category
3.The Ascoli- Arzela’ theorem
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. Frechet Space, Introduction to Theory of Distribution
Definitions and Examples, Operations on Distributions, Fourier Transform, Applications to Partial Differential Equations.
II.Course prerequisite：
Backgrounds: Advanced Calculus, Linear Algebra, and Function of Real Variables if possible
III.Reference material ( textbook(s) )：
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.
4. Douglas N. Arnold, Functional Analysis,http://www.math.psu.edu/dna/
IV.Grading scheme：
20% Exercises, 40% Mid- term Exam, 40% Final Exam