### Reminiscences

#### "My space"

Most my understanding of mathematics, I owe it to the 'Youth School of Mathematics', a Soviet voluntary system of math study groups for gifted secondary school children. The system was really excellent in the 60-th. The instructors were enthusiastic university students. One of such groups in Leningrad, contained Kharlamov, Kislyakov, Reyman and me, was instructed by Kruglov and Lifschitz (many thanks to both). There, I got used to the perplexity of a mathematician facing a problem, to the ordinary miracle of overcoming the perplexity, and to the ordinary sorrow when the miracle does not come. Kruglov taught mathematical analysis (from the definition of a metric space till spectral theory of operators in Hilbert spaces), while Lifschitz taught logic. (Is logic relevant to Banach spaces? Wait a little!)

While being a pupil in the 'youth school of math' (and a standard secondary school, of course), I have discovered for myself the (well-known to specialists long ago) fact that spaces lp are non-isomorphic to each other.

While being a graduate student, I was impressed by Cohen's 'forcing method'. I gave a long series of talks on a seminar (on Boolean algebras, D. Vladimirov), presenting Cohen's (1963) historic result (the continuum hypothesis does not follow from Zermelo-Frenkel axioms of set theory) by means of Boolean-valued models (following D.S. Scott and R. Solovay). However, I never worked in logic.

My post-graduate study was in probability theory, but that is another story. In parallel, during my first post-graduate year, I was thinking about a well-known problem: does every infinite-dimensional Banach space contain a subspace, linearly homeomorphic to one of lp or c0 ? Of course, I was acquainted with relevant papers by Milman.

Was I solving the problem? Really, not. Rather, I was staying perplexed. Good ideas did not come; the ideas that came were miserable. I believed in existence of a counterexample. Sometimes I felt an envy at Paul Cohen: he found his pioneering way of designing an appropriate model for the whole mathematics, whereas I cannot design an appropriate Banach space!

Well, Cohen constructs his object gradually, step-by-step. Each step depends on the current state (a feedback), and imposes new restrictions, both from above and from below, on the object under construction. What a contrast to us, Banach geometers! We define a space at once. Maybe, just maybe... can I construct a convex set (the unit ball of the future space) step-by-step? I could impose a restriction from below (on the ball; that is, from above on the norm) by adding a point to the ball, and from above - by adding a point to the dual ball. A feedback could ensure consistency and enforce the needed property. A constructive idea, isn't it?

The stubborn perplexity was thus overcome. You see, "the first truly nonclassical Banach space" was induced by the first truly non-Godelian model in logic.

The idea was thought over during several three-hours walks. Once, a pedestrian asked me about the way to LOMI (Leningrad branch of Steklov mathematical institute). I had explained him the way, and continued thinking. Only after a while I have realised the humor: a casual pedestrian, kilometers apart from LOMI, had a slim chance to know, what LOMI is, and where. Apparently, my face was showing that I am solving a mathematical problem...

We could choose a number p0, exclude all lp with p < p0 by adding points (linear combinations of already present finite block-sequences) to the ball, and exclude all p > p0 and the very p0 by adding points to the dual ball. Though, maybe, it is easier to move p0 to infinity. Exclude all lp by adding points to the ball, and c0 by reflexivity. To this end, the ball should be kept compact in c0. So, we combine blocks almost as freely as in c0; can the 'almost' save the compactness? The ball must be a huge compact set in c0. Why not? Such sets can be constructed easily. And so, the problem is solved!

I have given a talk on an extraordinary seminar in Leningrad university, and have written a paper. Now, the referee got perplexed. I was quite young, and all my experience in Banach spaces was a righteously rejected, poor manuscript about some techniques for calculating Banach-Mazur distances. Feeling that a mistake must exist, but unable to find any mistake during two weeks, the referee has recommended my work to publication.

Suddenly, the publication was stopped by 'competent bodies' (a Soviet euphemism for KGB, Party and State control, censorship, and all that) because of mentioning a poor, obscene name, though, not long ago a well-known Soviet mathematician, but now a contemptible apostate and despicable turncoat, wishing to emigrate from Soviet Union to Israel... Did you guess the name? Yes, it was Vitali Milman. Alas, the (quite natural) reference to him was deleted by me; otherwise the paper could not be published in Soviet Union. As I understand now, it could be published abroad, unlawfully; but in 1973 I was far from such ideas.

The paper, appeared in 1974, is my sole paper on Banach spaces geometry. Several weeks after solving the problem I was feeling able to solve many other problems readily. Having no other way of driving away the absurd feeling, I tackled the Banach's hyperplane problem, which was quite effective: the feeling have vanished quickly. The new problem required a new bright idea; the Cohen's manna from heaven was exhausted. Fearing to be shadowed by my own past, I have preferred to leave Banach spaces for probability. Interestingly, one of corner-stones of Gowers' (1994) solution of the hyperplane problem is just my (1974) construction, which was quite unexpected for me.

Forcing, and Boolean-valued models
The following nice books are available to you now (but not to me in 1970):
• H.G. Dales, W.H. Woodin, "An introduction to independence for analysts." Cambridge Univ. Press, 1987. [90d:03101]
• Thomas J. Jech, "The axiom of choice." North-Holland publ. co., 1973. [53#139]
Relevant papers by Milman
See especially "The fundamental structure conjecture" on page 145 of: V.D. Mil'man, "Geometric theory of Banach spaces. Part II: Geometry of the unit sphere", Russian Mathematical Surveys 26:6 (1971), 79-163 (translated from Russian).
Kharlamov
Viatcheslav M. Kharlamov. See for instance: "Real algebraic surfaces" (in Russian), Proc. Intern. Congress Math. (Helsinki 1978), 421-428. [82e:14032]
Kislyakov
Serguei V. Kislyakov. See for instance: "A sharp correction theorem", Studia Math. 113:2 (1995), 177-196. [96j:42005]
Reyman
Alex G. Reyman. See for instance: "The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions" (with Bobenko A.I. and Semenov-Tian-Shansky M.A.), Commun. Math. Phys. 122:2 (1989), 321-354. [91a:58070]
Kruglov
Alexander A. Kruglov. See for instance: "Certain spaces of means" (in Russian), Vestnik Leningrad. Univ. 1972, no. 1, 155-156. [45#5742]
Lifschitz
Vladimir Lifschitz. See for instance: "Semantical completeness theorems in logic and algebra", Proc. Amer. Math. Soc. 79:1 (1980), 89-96. [81g:03073]