Arnold D.N. Functional Analysis

Department of Mathematics, Penn State University, 1997. — 36 pages. OCR.These lecture notes were prepared for the instructor's personal use in teaching a half-semester course on functional analysis at the beginning graduate level at Penn State, in Spring 1997.
Vector spaces and their topology.
Subspaces and quotient spaces.
Basic properties of Hilbert spaces.
Linear Operators and Functionals.
The Hahn-Banach Theorem.
Duality.
Fundamental Theorems.
The Open Mapping Theorem.
The Uniform Boundedness Principle.
The Closed Range Theorem.
Weak Topologies.
The weak topology.
The weak* topology.
Compact Operators and their Spectra.
Hilbert-Schmidt operators.
Compact operators.
Spectral Theorem for compact self-adjoint operators.
The spectrum of a general compact operator.
Introduction to General Spectral Theory.
The spectrum and resolvent in a Banach algebra.
Spectral Theorem for bounded self-adjoint operators.