Linear Methods

Linear and normed spaces. Completeness, Banach spaces and Banach's fixed point theorem. Picard's theorem. Linear transformations. Inner product spaces, projections, and Hilbert spaces. Orthogonal sequences and approximations. Linear functionals, dual space, and Riesz' representation theorem. Spectral theorem, Jordan canonical form, and matrix decompositions.

1. Knowledge. The student has knowledge of central concepts in the theory of vector spaces, normed spaces and Hilbert spaces. In the theory of vector spaces the main objective is that the student understand the transition from Euclidean spaces to general vector spaces. This includes an understanding of isomorphisms and bases of finite dimensional vector spaces and the relationship between linear transformations and matrices. The student is familiar with principles of matrix factorization. In the theory of normed spaces a key objective is that the student understand the Banach fixed point theorem. This includes an understanding of convergence of sequences and continuous functions. The student masters the basic concepts from the theory of Hilbert spaces, including orthogonality, closest point and duality. The student understands the Riesz representation theorem.

2. Skills. The student is able to apply his or her knowledge of the theory of vector spaces, normed spaces and Hilbert spaces to solve concrete problems. A key skill is that the student is able to combine results and construct new proofs using the theory acquired in the course.

Lectures and mandatory exercises.