# Combinatorics in Banach Space Theory (University of Warsaw 2014/15)

Here you may download the lecture notes in Polish (which are still to be updated/corrected):

• Notes (uploaded on 18.01.2015)

# Combinatorics in Banach Space Theory (University of Silesia 2012/13)

• Combinatorics in Banach Space Theory 1. Almost disjoint families and Rosenthal's lemma with their applications: Phillips' lemma, Schur's property of $$\ell_1$$, the structure of weakly compact operators on injective Banach spaces, non-injectivity of $$\ell_\infty/c_0$$. Basic properties of weakly compactly generated ($$\mathsf{WCG}$$) Banach spaces. The Johnson-Lindenstrauss space as an example showing that being $$\mathsf{WCG}$$ is not a three-space property. A quantitative version of Krein's theorem on weak compactness of the closed convex hull of a weakly compact set, via Pták's combinatorial lemma. Steinitz's lemma and Lyapunov's theorem on the range of a vector measure; $$B$$-convex spaces; Kalton-Giesy theorem: $$B$$-convexity is a three-space property. Rudiments of the three-space problem; $$K$$-spaces; Kalton-Roberts theorem on nearly additive set functions; $$c_0$$ and $$\ell_\infty$$ are $$K$$-spaces.

• Combinatorics in Banach Space Theory 2. Basics of Ramsey theory: the Ellentuck topology, completely Ramsey sets, the Galvin-Prikry theorem, and their applications in the proof of Rosenthal's $$\ell_1$$-theorem. Elements of the theory of bases and basic sequences in Banach spaces; the Bessaga-Pe³czyñski selection principle. The Erdös-Magidor dichotomy and its consequence: the closedness of the Banach-Saks operator ideal. Construction of the Tsirelson space, i.e. a separable reflexive Banach space with no copy of $$\ell_p$$ ($$1\leq p<\infty$$) and $$c_0$$. Enflo's construction of a separable reflexive Banach space failing the approximation property (and thus not having a Schauder basis). The (sketch of the) Gowers-Maurey solution of the unconditional basic sequence problem.

Here you may download the lecture notes and some accompanying files.