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# Mathematics: Linear Functional Analysis

Course · 7.5 credits

## Description

The course presents the principles of functional analysis and their applications. These include the basic properties of Banach and Hilbert spaces as well as the spectral theory of bounded and compact linear operators.

The course treats fundamental properties of Banach and Hilbert spaces and the bounded linear operators defined on them:

• Banach spaces, the Hahn-Banach Theorem, weak convergence and  weak precompactness of the unit ball.
• Hilbert spaces. Examples including L2 spaces. Orthogonality, orthogonal complement, closed subspaces, projection theorem. Riesz Representation Theorem.
• Orthonormal sets, Bessel's inequality. Complete orthonormal sets, Parseval's identity.
• Baire Category Theorem and its consequences for operators on Banach spaces (uniform boundedness, open mapping, inverse mapping and closed graph theorems). Strong convergence of sequences of operators.
• Bounded and compact linear operators on Banach spaces and their spectra.
• The spectral theorem for compact self-adjoint operators on Hilbert spaces.

## Course documents

Syllabus (PDF - new window)

## Contact

Mathematics (Faculty of Sciences)

Sölvegatan 18A, 223 62 Lund

Box 118, 221 00 Lund

## Requirements and selection

### 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 %.

### English language requirements

Most of Lund University’s programmes require English Level 6 (unless otherwise stated under 'Entry requirements'). This is the equivalent of an overall IELTS score of 6.5 or a TOEFL score of 90. There are several ways to prove your English language proficiency – check which proof is accepted at the University Admissions in Sweden website. All students must prove they meet English language requirements by the deadline, in order to be considered for admission.

### Country-specific requirements

Check if there are any country-specific eligibility rules for you to study Bachelor's or Master's studies in Sweden:

Country-specific requirements for Bachelor's studies – universityadmissions.se

Country-specific requirements for Master's studies – universityadmissions.se

## Apply

### Start Autumn Semester 2024

Day-time Lund, part time 25%

In English

Study period

2 September 2024 - 19 January 2025

Application

You can only apply for this course in the 'Swedish student' application round. Find out more: Applying for studies – when to apply

### How to apply

Lund University uses a national application system run by University Admissions in Sweden. It is only possible to apply during the application periods.

When to apply for studies

### Step 1: Apply online

• Check that you meet the entry requirements of the programme or course you are interested in (refer to the section above on this webpage).
• Start your application – go to the University Admissions in Sweden website where you create an account and select programmes/courses during the application period.
Visit the University Admissions in Sweden website
• Rank your programme/course choices in order of preference and submit them before the application deadline.

### Step 2: Submit documents

Country-specific requirements for Bachelor's studies – universityadmissions.se
Country-specific requirements for Master's studies – universityadmissions.se

- official transcripts and high school diploma (Bachelor's applicants)
- official transcripts and degree certificate or proof that you are in the final year of your Bachelor's (Master's applicants)
- passport/ID (all applicants) and
- proof of English proficiency (all applicants).

• Prepare programme-specific documents if stated in the next paragraph on this webpage.
• Pay the application fee (if applicable – refer to the section below on this webpage) before the document deadline.

* Note that the process is different if you are applying as an exchange student or as a part of a cooperation programme (such as Erasmus+).
* If you have studied your entire Bachelor's programme in Sweden and all of your academic credits are in Ladok, you do not have to submit transcripts or your diploma when applying for a Master's programme. However, there may still be other documents you need to submit! See the link below.

## Tuition fees

### Non-EU/EEA citizens

Full programme/course tuition fee: SEK 21 250

First payment: SEK 21 250

Convert currency – xe.com

Citizens of a country outside of the European Union (EU), the European Economic Area (EEA) and Switzerland are required to pay tuition fees. You pay one instalment of the tuition fee in advance of each semester.

Tuition fees, payments and exemptions

### EU/EEA citizens and Switzerland

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

### Application fee

If you are required to pay tuition fees, you are generally also required to pay an application fee of SEK 900 when you apply at the University Admissions in Sweden website. You pay one application fee regardless of how many programmes or courses you apply to.

*Note that there are no tuition or application fees for exchange students or doctoral/PhD students, regardless of their nationality.

## Scholarships & funding

### Lund University Global Scholarship programme

The Lund University Global Scholarship programme is a merit-based and selective scholarship targeted at top academic students from countries outside the EU/EEA.

Lund University Global Scholarship

### Swedish Institute Scholarships

The Swedish Institute offers scholarships to international students applying for studies in Sweden at Master's level.

Scholarship information on the Swedish Institute website

### Country-specific scholarships and funding options

Lund University has agreements with scholarship organisations and funding bodies in different countries, which may allow applicants to apply for funding or scholarships in their home countries for their studies at Lund University.

### External scholarships

Information about scholarships from external organisations