**Math 581-1 Functional Analysis 4(3+1)**

Banach spaces: Basic properties and examples, convex sets, subspaces and quotient spaces, linear functionals and the dual spaces, Hahn-Banach theorem, the uniform boundedness principle, the open mapping theorem and closed graph theorem. Hilbert spaces: the Riesz representation theorem, orthonormal bases, isomorphic Hilbert spaces. Operators on Hilbert spaces: Basic properties and examples, adjoints, projections, invariant and reducing subspaces, positive operators and the polar decomposition, self-adjoint operators, normal operators, isometric and unitary operators.