Hilbert Space: Midterm Review

Outline

Chapter 1

1. Vector spaces: Basis, linear independence
2. Normed spaces: Norm, convergence, equivalent norms
3. Topology of normed spaces: Open, closed, bounded, compact, and dense sets
4. Banach spaces: Cauchy sequences, completeness, convergent and absolutely convergent series
5. Linear mappings: Continuity, boundedness, the norm of a linear mapping

Chapter 3

1. Inner product spaces: Inner product, norm, Schwarz’s Inequality, Parallelogram Law, orthogonality, Pythagorean Formula, strong and weak convergence, Hilbert spaces, isomorphic Hilbert spaces
2. Orthonomal systems: Orthogonal and orthonormal systems, Gram-Schmidt orthonormalization process, Pythagorean Formula, Bessel’s equality and inequality, the best approximation, complete orthonormal sequences and equivalent conditions, Parseval’s Formula, separable Hilbert spaces
3. Orthogonal Complements and Projections: Orthogonal sets, orthogonal complement, the closest point property, orthogonal decompositions, orthogonal projections
4. Linear functionals: The norm of a linear functional, the Riesz Representation Theorem