CMS/ACM 107 Introduction to Linear Analysis with Applications

Covers the basic algebraic, geometric, and topological properties of normed linear spaces, inner-product spaces, and linear maps. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.

  • Finite dimensions: norms and inner-products on vectors and matrices; norm equivalence; eigenvalues and eigenvectors, matrix factorizations including normal diagonalization, Jordan form, Cholesky and the singular value decomposition; four subspace theorem; pseudo-inverse.
  • Banach and Hilbert spaces: linear operators, dual spaces, norms and inner-products, convexity, closest point, orthogonality, projection and least squares. lp spaces, density, Schauder bases and separability. Continuous and compact embeddings. Sequential compactness. Reisz representation. Lax-Milgram.
  • Functional Analysis: Banach algebras, contraction mapping principle, boundedness and continuity of linear maps, inverses of linear maps and Neumann series, adjoints. Spectral theorem for compact symmetric operators. Functional calculus. Topological dual of a normed space. Hahn-Banach theorem and consequences.
  • Function Spaces: continuous and differentiable functions, extreme value and boundedness theorems, Lp spaces, Fourier transform, Sobolev spaces.