Topological vector spaces and applications to geometry. Fall 2017

A. Yu. Pirkovskii

Syllabus

Lecture 1 (21.09.2017).

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, absolutley convex sets and hulls.
Exercises for Lecture 1

Lecture 2 (05.10.2017).

Absorbing sets in vector spaces. The Minkowski functional. Locally convex spaces and their "polynormability". Continuity criteria for seminorms and linear maps. Extension of linear functionals from subspaces of locally convex spaces.
Exercises for Lecture 2

Lecture 3 (12.10.2017).

Equivalent families of seminorms. Examples. The strongest and the weakest locally convex topologies. Finite-dimensional, normable, and metrizable locally convex spaces.
Exercises for Lecture 3

Lecture 4 (19.10.2017).

Bounded sets in topological vector spaces. Basic properties and characterizations of bounded sets. Kolmogorov's normability criterion. Bounded sets and linear maps. Bornological locally convex spaces. Metrizable spaces are bornological. Characterizations of bornological locally convex spaces. Abstract bornological vector spaces and their relation to locally convex spaces. Topologically injective, open, and strict linear maps.
Exercises for Lecture 4

Lecture 5 (26.10.2017).

Quotients. The universal property of quotients. The associated Hausdorff space. Kernels and cokernels in the category of locally convex spaces. Projective locally convex topologies.
Exercises for Lecture 5

Lecture 6 (02.11.2017).

Inductive locally convex topologies. Examples: the spaces of continuous and smooth compactly supported functions, the spaces of holomorphic germs. Products and coproducts of locally convex spaces.

Lecture 7 (09.11.2017).

Limits and colimits of locally convex spaces. Examples.

Lecture 8 (16.11.2017).

Topologies on spaces of linear maps. Special cases: the strong dual and the weak dual. 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. The completeness of spaces of linear maps.

Lecture 9 (23.11.2017).

Examples of complete locally convex spaces. The completion of a locally convex space. The completion as the projective limit of the associated Banach spaces. The completion as a functor. The projective tensor product of seminormed spaces.

Lecture 10 (30.11.2017).

The projective tensor product of seminormed spaces: the universal property and an explicit construction. Injective tensor product of seminormed spaces. Reasonable cross-seminorms on tensor products. The maximality of the projective tensor seminorm and the minimality of the injective tensor seminorm in the class of reasonable cross-seminorms. The projective tensor product of locally convex spaces. The functoriality of the projective tensor product.

Lecture 11 (07.12.2017).

The injective tensor product of locally convex spaces. The functoriality of the injective tensor product. Some canonical isomorphisms related to the projective tensor product. Projective and injective tensor products of Hausdorff saces are Hausdorff. Cokernels in the categories of complete locally convex spaces and of seminormed spaces. The projective tensor product preserves cokernels.

Lecture 12 (14.12.2017).

The uncompleted injective tensor product preserves kernels. Projective tensor product of L^1-spaces, of Köthe spaces, of the spaces of smooth functions on tori, of the spaces of holomorphic functions on polydiscs. Injective tensor products of spaces of continuous functions.

Lecture 13 (18.12.2017).

Nuclear maps between Banach spaces. Equivalent definitions and basic properties of nuclear maps. Examples (diagonal operators, integral operators). Factorization of nuclear maps through Hilbert spaces. Nuclear maps between locally convex spaces. Banach disks and factorization of nuclear maps through the associated Banach spaces. Nuclear locally convex spaces. The Grothendieck-Pietsch nuclearity criterion for Köthe spaces. The stability of nuclearity under standard constructions. Examples of nuclear spaces (the space of rapidly decreasing sequences, spaces of smooth and holomorphic functions, the Schwartz space). The equality of the projective and injective tensor products by a nuclear space.

Lecture 14 (21.12.2017).

Sheaf cohomology. The Čech complex and Leray's theorem. Coherent analytic sheaves. H. Cartan's "Theorem B". The canonical topology on 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.