Topological vector spaces. Spring 2026

A. Yu. Pirkovskii

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.

Lecture 5 (11.02.2026) (video)

Projective (initial) topologies. Examples. Products of topological vector spaces. A category-theoretic approach to products. Projective limits in categories. Projective limits of topological vector spaces. Examples.

Lecture 6 (18.02.2026) (video)

Inductive (final) topologies: the inductive vector topology and the inductive locally convex topology. The existence and elementary properteis of inductive topologies. Continuous seminorms for the inductive locally convex topology. Examples: quotients, the strongest locally convex topology, the canonical topologies on spaces of continuous and smooth compactly supported functions and on spaces of holomorphic germs. Locally convex direct sums. A category-theoretic interpretation: coproducts. A defining family of seminorms on a locally convex direct sum. Inductive topologies as quotients of locally convex direct sums.

Lecture 7 (25.02.2026) (video)

Inductive limits in categories. A special case: inductive limits of directed families of subobjects. Inductive limits of vector spaces and of locally convex spaces. Inductive systems of vector spaces with injective connecting maps = directed families of subspaces. Examples of locally convex inductive limits. Strict inductive limits. Examples. Properties of strict countable inductive limits.

Exercise sheets

• Exercise sheet 1. Topological vector spaces. Seminorms and convexity
• Exercise sheet 2. Continuous linear operators. Equivalent families of seminorms
• Exercise sheet 3. Normable and metrizable locally convex spaces
• Exercise sheet 4. Bounded sets
• Exercise sheet 5. Quotients, kernels, cokernels
• Exercise sheet 6. Projective topologies. Products and projective limits

Earlier courses

• Fall 2023
• Fall 2017
• Fall 2015 (in Russian)
• Spring 2013 (in Russian)
• Spring 2007 (in Russian)