Topological vector spaces. Spring 2026
Syllabus
Lectures
Playlist
- Lecture 1 (14.01.2026) (video)
- Historical remarks on topological vector spaces.
Topological vector spaces. The topology generated by a family of seminorms.
Examples: spaces of continuous, smooth, holomorphic functions, the Schwartz space,
the space of rapidly decreasing sequences, the weak and the weak* topologies.
Convex, circled, absolutely convex sets and hulls. Absorbing sets in vector spaces. The Minkowski
functional.
- Lecture 2 (21.01.2026) (video)
- Basic properties of the Minkowski functional. Locally convex spaces (l.c.s's) and their "polynormability".
Continuity criteria for seminorms and linear maps. Examples of continuous linear operators between l.c.s's
(differential operators on spaces of smooth and holomorohic functions,
the Fourier transform of smooth functions). Extension of linear functionals from subspaces of l.c.s's.
Linear functionals on a Hausdorff l.c.s. separate the points.
- Lecture 3 (28.01.2026) (video)
- The domination relation and equivalence for families of seminorms. Examples of equivalent families of seminorms.
The strongest and the weakest locally convex topologies. Directed families of seminorms. Classification of finite-dimensional l.c.s's.
A normability criterion for l.c.s's in terms of seminorms.
F-seminorms. A metrizability criterion for l.c.s's.
- Lecture 4 (04.02.2026) (video)
- Bounded sets in topological vector spaces. Characterizations and basic properties of bounded sets.
Remarks on abstract bornological vector spaces. Kolmogorov's normability criterion. Bounded sets and linear maps.
Bornological locally convex spaces. Metrizable spaces are bornological. Characterizations of
bornological locally convex spaces. Quotients of topological vector spaces. Basic properties of quotients.
Quotient seminorms. The universal property of quotients.
Exercise sheets
References
Earlier courses