Topological vector spaces. Fall 2023

A. Yu. Pirkovskii

Syllabus

Lecture 1 (08.09.2023) (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 (15.09.2023) (video)

Locally convex spaces (l.c.s's) and their "polynormability". Continuity criteria for seminorms and linear maps. Extension of linear functionals from subspaces of l.c.s's. Linear functionals on a Hausdorff l.c.s. separate the points.

Lecture 3 (22.09.2023) (video)

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). 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. Bounded subsets of a topological vector space. Kolmogorov's normability criterion. F-seminorms. A metrizability criterion for l.c.s's.

Lecture 4 (29.09.2023) (video)

Quotients of topological vector spaces. The universal property of quotients. Kernels and cokernels in categories of topologial vector spaces. Projective and inductive locally convex topologies. Examples. The canonical topologies on the spaces of continuous and smooth compactly supported functions and on the spaces of holomorphic germs.

Lecture 5 (06.10.2023) (video)

Products of locally convex spaces. Coproducts (locally convex direct sums) of locally convex spaces. Limits of locally convex spaces. Examples.

Lecture 6 (13.10.2023) (video)

Limits and colimits of locally convex spaces. Examples.

Lecture 7 (20.10.2023) (video)

Strict inductive limits. Examples. Properties of strict countable inductive limits. Topologies of uniform convergence on spaces of linear maps. Special cases: the strong dual and the weak dual.

Lecture 8 (27.10.2023) (video)

Convergence of nets in topological spaces. Complete topological vector spaces. The "extension by continuity" theorem. Completeness vs. metric completeness for metrizable spaces. Fréchet spaces. Completeness and constructions (products, limits, coproducts). The completeness of quotients of Fréchet spaces. Examples of complete locally convex spaces. The completion of a locally convex space (definition, uniqueness, and a reduction to the Hausdorff case).

Lecture 9 (03.11.2023) (video)

The completion of a locally convex space as the projective limit of the associated Banach spaces. The completion as a functor. The completeness of strict inductive limits. Basic facts on bilinear maps between locally convex and seminormed spaces. The projective tensor product of seminormed spaces: the universal property and an explicit construction.

Lecture 10 (10.11.2023) (video)

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. The projective tensor product and the complete projective tensor product of locally convex spaces. The functoriality of the projective tensor product.

Lecture 11 (17.11.2023) (video)

The injective tensor product of locally convex spaces. Comparing with the projective tensor product. The functoriality of the injective tensor product. The projective and injective tensor products of Hausdorff spaces are Hausdorff. Some canonical isomorphisms involving projective and injective tensor products. Some explicit calculations of projective and injective tensor norms. The projective tensor product of L¹-spaces. Köthe sequence spaces. Examples.

Lecture 12 (24.11.2023) (video)

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. The projective tensor product preserves cokernels and (as a corollary) quotients.

Lecture 13 (01.12.2023) (video)

The injective tensor product preserves topological and isometric embeddings. A series decomposition for elements of the complete projective tesnor product of metrizable spaces. Nuclear maps between Banach spaces. Equivalent definitions and basic properties of nuclear maps. Examples (diagonal operators, integral operators). Nuclear maps between locally convex spaces. Banach disks. Equicontinuous families of linear maps. Factorization of nuclear maps through the associated Banach spaces.

Lecture 14 (08.12.2023) (video)

Factorization of nuclear maps through Hilbert spaces. The nuclearity of the projective tesnor product of nuclear maps. The nuclearity of restrictions and quotients of nuclear maps between Hilbert spaces. 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.

Lecture 15 (15.12.2023) (video)

More examples of nuclear spaces (spaces of smooth and holomorphic functions on manifolds, the Schwartz space, spaces of holomorphic germs). The equality of the projective and injective tensor products by a nuclear space. The complete projective and injective tensor products commute with reduced projective limits and (as a corollary) with products. The complete injective tensor product preserves injections.

Lecture 16 (22.12.2023) (video1, video2)

The exactness of the complete projective tensor product for nuclear spaces. The complete projective tensor product of the spaces of holomorphic (resp. smooth) functions on complex (resp. real) manifolds. The Čech complex and Leray's theorem. Coherent analytic sheaves on complex 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.

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. Quotients, kernels, cokernels
• Exercise sheet 5. Projective and inductive locally convex topologies. Products and coproducts
• Exercise sheet 6. Projective and inductive limits
• Exercise sheet 7. Completeness
• Exercise sheet 8. Köthe sequence spaces
• Exercise sheet 9. Topological tensor products
• Exercise sheet 10. Topological tensor products II
• Exercise sheet 11. Nuclear operators and nuclear spaces