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.
Lecture 8 (04.03.2026) (video)
Properties of strict countable inductive limits. Topologies of uniform convergence on spaces of linear maps. Special cases: the strong dual and the weak dual. Convergence of nets in topological spaces. Cauchy nets and completeness for topological vector spaces. Completeness vs. metric completeness for metrizable spaces. Fréchet spaces.
Lecture 9 (11.03.2026) (video)
The "extension by continuity" theorem. Completeness and constructions (products, projective limits, direct sums). The completeness of quotients of Fréchet spaces. The completeness of L(X,Y) where X is bornological and Y is complete. Examples of complete locally convex spaces. Bornological spaces and inductive topologies. Spaces of distributions and their completeness.
Lecture 10 (18.03.2026) (video)
The completion of a locally convex space: definition, uniqueness, and a reduction to the Hausdorff case. The completion of a locally convex space as the projective limit of the associated Banach spaces. The completeness of strict inductive limits. Equicontinuous families of linear maps. Equicontinuous families are bounded for every topology of uniform convergence on L(X,Y).
Lecture 11 (01.04.2026) (video)
Barrelled spaces. Fréchet spaces are barrelled. Barrelled spaces and inductive topologies. The Banach-Steinhaus theorem for barrelled spaces. Quasibarrelled spaces. Separately continuous bilinear maps on Fréchet spaces are continuous. The Banach-Alaoglu-Bourbaki theorem. The open mapping theorem and the closed graph theorem for Fréchet spaces.
Lecture 12 (08.04.2026) (video)
Basic facts on bilinear maps between seminormed spaces. The projective tensor product of seminormed spaces: the universal property and an explicit construction. Reasonable cross-seminorms on tensor products. The maximality of the projective tensor seminorm. The injective tensor product of seminormed spaces. The minimality of the injective tensor seminorm. An example: the projective tensor product by Kn1 and the injective tensor product by Kn∞.
Lecture 13 (15.04.2026) (video)
The projective tensor product and the complete projective tensor product of locally convex spaces: universal properties, construction, functoriality. The injective tensor product of locally convex spaces: construction, functoriality. Comparing with the projective tensor product. The projective and injective tensor products of Hausdorff spaces are Hausdorff. Some canonical isomorphisms involving projective and injective tensor products. The projective tensor product of L1-spaces.
Lecture 14 (22.04.2026) (video)
Köthe sequence spaces. Examples. The projective tensor product of Köthe spaces. Corollaries: the projective tesnor products of the spaces of smooth functions on tori and of the spaces of holomorphic functions on polydiscs. Calculating the injective tensor seminorm via norming sets. The injective tensor products of spaces of continuous and smooth functions and of the Schwartz spaces. Kernels and cokernels in categories with 0. Characterization of kernels in categories of locally convex spaces.
Lecture 15 (29.04.2026) (video)
Characterization of cokernels in categories of locally convex spaces. The projective tensor product preserves cokernels and (as a corollary) quotients. The injective tensor product preserves topological and isometric embeddings. The complete projective and injective tensor products commute with reduced projective limits and (as a corollary) with products. A series expansion for elements of the complete projective tesnor product of metrizable spaces.
Lecture 16 (13.05.2026) (video)
Nuclear operators between Banach spaces. Equivalent definitions and basic properties of nuclear operators. Examples (diagonal operators, integral operators). Nuclear operators between locally convex spaces. Banach disks. Factorization of nuclear operators through the associated Banach spaces. Factorization of nuclear operators through Hilbert spaces. The nuclearity of the projective tensor product of nuclear operators. The nuclearity of restrictions and of quotients of nuclear operators between Hilbert spaces.
Lecture 17 (20.05.2026) (video)
Nuclear locally convex spaces. Charachterizations of nuclearity. The Grothendieck-Pietsch nuclearity criterion for Köthe spaces. Basic examples of nuclear spaces. The stability of nuclearity under standard constructions. More examples of nuclear spaces (spaces of smooth and holomorphic functions on manifolds, the Schwartz space, spaces of holomorphic germs).
Lecture 18 (27.05.2026) (video)
Montel and semi-Montel spaces. Complete nuclear spaces are semi-Montel. The equality of the projective and injective tensor products by a nuclear space. The complete injective tensor product preserves injections. The exactness of the complete projective tensor product for nuclear Fréchet spaces. The complete projective tensor product of the spaces of holomorphic (resp. smooth) functions on complex (resp. real) manifolds.
Lecture 19 (03.06.2026) (video)
The Čech complex and Leray's theorem. Coherent analytic sheaves on complex manifolds. Stein manifolds. The canonical topology on the section spaces of coherent analytic sheaves. A Schwartz-type theorem on nuclear perturbations. Binuclear maps. Binuclearly quasiisomorphic complexes of Fréchet spaces have finite-dimensional cohomology. The Cartan-Serre finiteness theorem.
Lecture 20 (10.06.2026) (video)
The Schwartz kernel theorem: an informal discussion. Distributions and operators with distribution kernels. The statement of the kernel theorem. Abstract kernel theorems for nuclear spaces. The nuclearity of the strong dual of a Fréchet space. The strong dual of a strict inductive limit. The nuclearity of the spaces of distributions. The adjoint associativity for Fréchet spaces. Grothendieck's theorem on bounded sets in the projective tensor product of nuclear Fréchet spaces. Corollaries: the topological adjoint associativity for nuclear Fréchet spaces; the strong dual of the complete tensor product of nuclear Fréchet spaces. A tensor product interpretation and the proof of the kernel theorem.

Exercise sheets

(Thanks for corrections to Zeling Ma and Andrei Goguadze)

References

Earlier courses