Combinatorics in Banach space theory

I presented a course entitled Combinatorics in Banach space theory twice: during the academic year 2012/13 at the University of Silesia in Katowice and during the winter semester of 2014/15 at the University of Warsaw. Those two courses differ quite significantly. The first one focused on several, more or less elementary, combinatorial principles and their various (often by no means elementary) applications to Banach space theory. It also contained some Banach-space-theoretical aspects of combinatorial geometry and applications to several three-space problems.

The second course starts with infinite Ramsey theory and then, via mixed Tsirelson spaces, proceeds to nowadays classical constructions of exotic Banach spaces. Most notably, we discuss the famous Gowers-Maurey construction of a hereditarily indecomposable (HI) Banach space, and also their general approach to the construction of Schlumprecht's space which yields a solution to the distorion problem. We analyze some structural properties of operators acting on complex HI spaces and, finally, Gowers' solution to the Schroeder-Bernstein problem for Banach spaces.

Below, there are lecture notes for both courses, the latter one in Polish. There are some supplementary problem sets devoted to each course which you can find below the list of lectures.

Course given at the University of Silesia (2012/13)

Summary. 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 (WCG) Banach spaces. The Johnson-Lindenstrauss space as an example showing that being 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.

• Notation
• Lecture 1
  Uncountable almost disjoint families and the Phillips-Sonczyk theorem on noncomplementability of \(c_0\) in \(\ell_\infty\). Rosenthal's lemma on diagonalization procedure for families of uniformly bounded, finitely additive measures.
• Lecture 2
  Semivariation and variation of vector measures. Nikodým's boundedness principle and Phillips' lemma on pointwise convergent sequences of measures. Schur's property of \(\ell_1\). The dual space \(\ell_\infty^\ast\) as the space of finitely additive measures on \(\mathcal{P}\mathbb{N}\) with the variation norm.
• Lecture 3
  Announcement of two Lindenstrauss theorems on the structure of \(\ell_\infty(\Gamma)\)-spaces. Equi-integrable subsets of \(L_1(\mu)\) and the Dunford-Pettis characterization of weak compactness in \(L_1(\mu)\). Grothendieck's characterization of weak compactness in the space \(\mathcal{M}(K)\) of regular Borel measures.
• Lecture 4
  Weakly compact operators and Gantmacher's theorem. Pełczyński's characterization of weakly compact operators on \(C(K)\)-spaces, with the use of Rosenthal's lemma and Grothendieck's theorem on relatively weakly compact sets of measures. Rosenthal's theorem on operators which are bounded below on a subspace isomorphic to \(c_0(\Gamma)\). An application to another Rosenthal's result on non-weakly compact operators on injective Banach spaces. Two important corollaries: the Bessaga-Pełczyński theorem and the Amir theorem on non-injectivity of \(\ell_\infty/c_0\).
• Lecture 5
  Direct sums and the Pełczyński decomposition method. Lindenstrauss' theorem on complemented subspaces of \(\ell_\infty(\Gamma)\) which contain \(c_0(\Gamma)\). The Dunford-Pettis property of Banach spaces; completion of the proof of the Dunford-Pettis characterization of weakly compact subsets of \(L_1(\mu)\). Grothendieck's theorem that \(C(K)\)-spaces enjoy the Dunford-Pettis property. Proof of the Lindenstrauss theorem: \(\ell_\infty\) is prime.
• Lecture 6
  Characterization of Grothendieck spaces; Grothendieck's theorem: \(C(K)\)-spaces with \(K\) being extremally disconnected are Grothendieck. Recalling basic facts about the Stone space of a Boolean algebra; ultrafilters and Stone-Čech compactifications. An example of \(C(K)\) which is not a Grothendieck space, yet \(K\) does not contain nontrivial convergent sequences. Dedekind complete Boolean algebras and \(\sigma\)-Stonean spaces. Haydon's theorem on the Grothendieck property of \(B(\mathcal{F})\) where \(\mathcal{F}\) has SCP property. Schachermayer's theorem: \(B(\mathcal{F})\) is not a Grothendieck space if \(\mathcal{F}\) is the union of a strictly increasing sequence of set algebras.
• Lecture 7
  Khintchine's inequality, both for real and complex scalars. A remark on optimal constants found by Haagerup. Rosenthal's theorem representing \(\ell_2(2^\Gamma)\) as a quotient of \(\ell_\infty(\Gamma)\).
• Lecture 8
  Weakly compactly generated spaces and canonical examples of such. General framework of the three-space problem. The Johnson-Lindenstrauss construction \(\mathrm{JL}_2\) and a resulting short exact sequence. Identification of the dual \(\mathrm{JL}_2^\ast\) and the conlusion that the dual being WCG does not imply that the original space is WCG. Being WCG is not a three-space property either.
• Lecture 9
  \(\varepsilon\)-relatively weakly compact sets and a result by Fabian, Hájek, Montesinos and Zizler about \(2\varepsilon\)-relative weak compactness of the convex hull. Perturbations of double-limit Grothendieck's characterization of relatively weakly compact sets. Upper weak\(^\ast\) semicontinuous envelopes. Pták's combinatorial principle, in a hereditary and nonhereditary version.
• Lecture 10
  Steinitz's combinatorial lemma presented after Bárány and Grinberg. The Darboux property of nonatomic scalar measures and the notion of atom for vector-valued measures. Recalling original Lyapunov's convexity theorem, the Bartle-Dunford-Schwartz theorem on relative weak compactness of the range of a \(\sigma\)-additive vector measure, as well as the Knowles theorem on the control measure and characterization of when a vector analogue of Lyapunov's theorem is true. A generalization of Lyapunov's theorem for almost atomless measures due to V. Kadets.
• Lecture 11
  Closedness of the range on any \(\sigma\)-additive vector measure with values in a finite-dimensional space. Uhl's example of a \(\sigma\)-additive measure with values in \(\ell_2\) and non-closed range. B-convex spaces in terms of 'Steinitz's constants'. Finite representability and an announcement of a characterization of B-convexity by non-admitting \(\ell_1\) as a finitely representable subspace. Submultiplicativity properties of Steinitz's constants. A mention about the Lindenstrauss-Pełczyński classes \(\mathscr{L}_p\).
• Lecture 12
  A crucial lemma about a connection between the Steinitz's constants and two other sequences involving the expectation values \(\mathbb{E}\|\sum \varepsilon_ix_i\|^2\). Finalization of the proof that \(X\) is B-convex if and only if \(\ell_1\) is not finitely representable in \(X\). Remarks on Beck's result that \(X\) is B-convex iff the strong law of large numbers holds true for \(X\)-valued random variables. V. Kadets' theorem on the convexity of the range of a \(\sigma\)-additive nonatomic vector measure of bounded variation taking values in a B-convex space. A mention on James' example of a nonreflexive B-convex Banach space. Introduction to the three-space problem for B-convexity: the Roelcke-Dierolf result that both being an F-space and being locally bounded are three-space properties in the category of linear topological spaces. Quasi-norms, p-norms and the Aoki-Rolewicz theorem. Kalton's characterization of local convexity.
• Lecture 13
  Kalton's sufficient condition for being isomorphic to a B-convex space. The Kalton-Giesy theorem saying that being isomorphic to a B-convex space is a three-space property in the category of quasi-normed spaces. Minimal extension and the notion of K-space. Twisted sums, equivalence, splitting and its characterizations in terms of retractions and liftings. Quasi-linear and zero-linear maps. The Kalton-Ribe-Roberts example of a non-locally convex twisted sum of \(\mathbb{R}\) and \(\ell_1\). Kalton's theorem which connects splitting of twisted sums with stability properties of quasi-linear maps. Posing the question: are \(c_0\) and \(\ell_\infty\) K-spaces?
• Lecture 14
  The formulation of the Kalton-Roberts theorem on nearly additive set functions. Preparations for showing that in fact every (quotient of a) \(\mathscr{L}_\infty\)-space is a K-space. \(C_0(K)\)-spaces are Lindenstrauss spaces, i.e. \(\mathscr{L}_{\infty,1+\varepsilon}\)-spaces, for each \(\varepsilon>0\). Assuming the Kalton-Roberts theorem we derive stability properties for real quasi-linear maps on \(\ell_\infty(\Omega)\) and conclude that every \(\mathscr{L}_\infty\)-space is a K-space. Special graphs called (super)concentrators. Using Hall's marriage lemma in order to prove Pippenger's result on the existence of certain special concentrators.
• Lecture 15
  Submeasures; Kalton-Roberts lemma on estimates for submeasures in terms of some quantities connected with concentrators. The covering index of a family of sets and Kelley's theorem on the existence of measures on Boolean algebras majorized by a covering index. Completion of the proof of the Kalton-Roberts theorem on stability of nearly additive set functions.
• References

Problem sets

• Part 1
• Part 2
• Part 3
• Part 4
• Part 5

Course given at the University of Warsaw (2014/15)

• Lectures 1-8 (written down by P. Ohrysko)
  Classical Ramsey theorem and the Brunel-Sucheston theorem on spreading models. The Ellentuck topology; the Nash-Williams and Galvin-Prikry theorems (Ramsey theorem for infinite subsets of \(\mathbb{N}\)). Application: the Rosenthal-Dor \(\ell_1\)-theorem and its consequences; the Odell-Rosenthal theorem. Rudiments of the theory of bases and basic sequences in Banach spaces; the small perturbation principle and the Bessaga-Pełczyński selection theorem. Shrinking and boundedly complete bases; James' characterization of reflexivity. The Tsirelson (Figiel-Johnson) space and James' \(\ell_1\)-distortion theorem. Mixed Tsirelson spaces and the Argyros-Deliyanni theorem, Representing \(c_0\) and \(\ell_p\)'s as mixed Tsirelson spaces; Schlumprecht's space as a mixed Tsirelson space. Asymptotic biorthogonal systems and connections with unconditional basic sequences. Gowers-Maurey approach to the proof of the fact that Schlumprecht's space is arbitrarily distortable: lower \(f\)-estimates, \(\ell_{1+}^k\)-averages, rapidly increasing sequences (RIS).
• Lectures 9-11
  Construction of the Gowers-Maurey space: the 'Schlumprecht part' and the part involving special functionals. Upper estimate on the norm of the sum of a RIS. Criterion on estimating the norm of the sum of a RIS by using special sequences of functionals and verifying an appropriate estimate on intervals. Hereditarily indecomposable (HI) Banach spaces and their characterizations. The Gowers-Maurey space is HI. Bounded linear operators on HI spaces; infinitely singular values, every bounded linear operator \(T\) on a complex HI space is of the form \(T=S+\lambda I\) with \(S\) strictly singular. The 'shift lemma' and a result on the existence of a universal sequence of approximate eigenvectors.
• Lecture 12-13
  Gowers' solution of the Schroeder-Bernstein problem for Banach spaces; construction of a Banach space being isomorphic to the direct sum of three copies of itself but not isomorphic to its square. Using a homomorphism of certain algebras of operators and behavior of the Fredholm index under that homomorphism. (To be continued.)

Problem sets

• Part 1
• Part 2
• Part 3
• Part 4