Course content

Linear and metric 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.

Learning outcome

1. Knowledge. The student has knowledge of central concepts in the theory of metric spaces, vector spaces and Hilbert spaces. In the theory of metric spaces a key objective is that the student understand the Banach fixed point theorem. This includes an understanding of metric spaces, convergence of sequences and continuous functions. 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. 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 metric spaces, vector 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.

Learning methods and activities

Lectures and mandatory exercises. The lectures will be given in English if they are attended by students from the Master's Programme in Mathematics for International students. Retake of examination may be given as an oral examination.

Compulsory assignments

• Exercises

Recommended previous knowledge

The course is based on TMA4100/05/15/20 Calculus 1/2/3/4K, or equivalent. MA1101 Basic Calculus I, MA1102 Basic Calculus II, MA1103 Vector Calculus, MA1201 Linear Algebra and Geometry, MA1202 Linear Algebra with Applications, MA2105 Complex Function Theory with Differential Equations (see course descriptions 2013/14), or equivalent.

Course materials

Will be announced at the start of the course.

Credit reductions