Mathematics: Linear Functional Analysis

This course introduces key concepts in functional analysis, focusing on Banach and Hilbert spaces and the behaviour of continuous linear maps. You will learn how these abstract structures underpin much of modern analysis and its applications, as well as some fundamental theorems related to their structure.

Topics include

• Banach spaces: Hahn–Banach Theorem, duality, and weak-star compactness
• Hilbert spaces: L² spaces, orthogonality, projection theorem, and Riesz representation
• Operator theory: The principles of functional analysis: Uniform boundedness, open and inverse mapping, and closed graph theorems
• Spectral theory: Bounded and compact operators, spectral properties, and the spectral theorem for compact self-adjoint operators.

The course is alternative-compulsory within the Master’s Programme in Mathematics at Lund University and is also offered as a standalone course. It is given in parallel with MATP45 Specialisation Course in Linear Functional Analysis, offering the possibility to dive deeper into the subject. We strongly recommend students to take both courses during the same semester for a comprehensive understanding of functional analysis.

You will take part in lectures and encouraged to actively discuss both examples and methods of proof. The assessment includes a written exam (5 credits) and an oral exam (2.5 credits) at the end of the course. The oral exam is only given for students who pass the written exam. Students who do not pass the first written exam will be offered a resit shortly after. Adjustments to the examination format may be made for students with permanent disabilities, in consultation with Disability Support Services.

Functional analysis is a cornerstone of advanced mathematics and has wide applications in areas like differential equations, quantum mechanics, and signal processing. After completing the course, you’ll be well-prepared for further studies in mathematical analysis or related fields, as well as for roles that require strong analytical and theoretical skills—such as research, data science, or mathematical modelling.

Entry requirements

At least 90 credits in pure mathematics including knowledge corresponding to a course in Integration Theory, 7.5 credits (Lebesgue integral). English 6/English course B.

Selection criteria

Seats are allocated according to: ECTS (HPAV): 100 %.

Tuition fees for non-EU/EEA citizens

Citizens of countries outside:

• The European Union (EU)
• The European Economic Area (EEA) and
• Switzerland

are required to pay tuition fees. You pay an instalment of the tuition fee in advance of each
semester.

Tuition fees, payments and exemptions

Full programme/course tuition fee: SEK 23,125
First payment: SEK 23,125

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Note that you may also need to pay an application fee, or provide proof of exemption.

Application fee

No tuition fees for citizens of the EU, EEA and Switzerland

There are no tuition fees for citizens of the European Union (EU), the European Economic Area (EEA) and Switzerland.