We have not addressed certain important problems that remain unsolved after many years concerning the classical Banach spaces themselves.

(Q13) Let K be a compact metric space. Is every complemented sub-space of C(K) isomorphic to C(L) for some compact metric space L?

It is known that if K is uncountable then C(K) is isomorphic to C[0,1]. If if is countable then C(K) is isomorphic to C(ωωα) for some α < ω1. Every complemented subspace of c0 (isomorphic to C (ω)) is either finite dimensional or isomorphic to c0 ([Pel]). If X is complemented in C[0,1] and X* is nonseparable then X is isomorphic to C[0,1] [R6]. Every quotient of c0 embeds isomorphically into c0 but this does not hold in general for C(ωωα). A discussion of these and related results may be found in [A1, A2, A3, A4], [Gal, Ga2], [Bo2].

The isomorphism types of the complemented subspaces of L1[0,1] remain unclassified.

(Q14) Let X be a complemented (infinite dimensional) subspace of L1[0,1]. Is X isomorphic to L1 or l1?

Every X which is complemented in lp (1 ≤ p < ∞) or c0 is isomorphic to lp or c0. There are known to be uncountably many mutually nonisomorphic complemented subspaces of Lp[0,1] (1 < p < ∞, p ≠ 2) [BRS] and all are known to have a basis [JRZ]. These spaces have been classified as ℒp spaces ([LP], [LR]), provided they are not Hilbert spaces.

The most outstanding problems in the theory of infinite dimensional Banach spaces, those that were central to the study of the general structure of a Banach space, finally yielded their secrets in the 1990's. In this survey we shall discuss these problems and their solutions and more. For many years researchers have been aware of deep connections between both the theorems and ideas of logic and set theory and Banach space theory. We shall try to illuminate these connections as well.

For example the ideas of Ramsey theory played a key role in H. Rosenthal's magnificent l1-theorem in 1974 [R1]. But there is also a less direct connection with the Banach space question as to whether or not separable infinite dimensional Hilbert space, l2, is distortable. This is equivalent to the following approximate Ramsey problem. Let Sl2 = {x ∈ l2: ∥x∥ = 1} be the unit sphere of l2. Finitely color the sphere by colors C1,…, Ck and let ε > 0. Does there exist an i0 and an infinite dimensional closed linear subspace X of l2 so that the unit sphere of X, SX, is a subset of (Ci0)ε = {y ∈ Sl2 : ∥y − x∥ < ε for some x ∈ Ci0}? It suffices to let (ei) be an orthonormal basis for l2 and confine the search to block subspaces — those spanned by block bases of (ei) (these terms are defined precisely below).

The articles in this book had their origins in three mini-courses offered at the conference “Analyse & Logique” held August 25–29, 1997, at the University of Mons-Hainaut in Möns, Belgium. For a long time there have been rich connections between analysis and logic; these articles bear witness that this relationship is still very active, and continues to be important for both areas.

Here we briefly describe these three articles; each one has a more detailed Introduction as its first chapter.

Part One: Ultraproducts in Analysis by C. Ward Henson and Jose Iovino

Applications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine, and of nonstandard hulls by Luxemburg. These constructions have been widely and successfully used in many parts of analysis. This paper presents the basic aspects of a systematic model theoretic framework within which these tools are naturally situated.

The logic developed here has its origin in the question: "what does a normed space structure have in common with its Banach space ultrapowers?" To give a precise answer it is necessary to introduce a suitable formal language of positive bounded formulas together with a semantics of approximate satisfaction. The resulting theory is developed here for a very general class of structures based on normed spaces; our treatment has the most general context possible within functional analysis in which the Banach space ultraproduct and nonstandard hull constructions apply.

In this chapter we prove isomorphism theorems for ultrapowers and ultraproducts of normed space structures. These results show that there is a very tight connection between (a) properties that are preserved under the ultraproduct construction and (b) properties that are expressible using the logic for normed space structures that is described in this paper.

Let L be a signature and let ℳ and N be two normed space L-structures. If ℳ and N have isomorphic ultrapowers, by Corollary 9.4 they must be approximately elementarily equivalent. Theorem 10.7 below gives the converse (in a strong form). Together these results show that ultrapower equivalence of ℳ and N is the same as approximate elementary equivalence. (See the discussion of this issue in the Introduction.)

Theorem 10.8 below is a similar result for ultraproducts. Among other things, it shows that the ultrafilter guaranteed by Theorem 10.7 can be chosen in a highly uniform way and that the ultrapowers in question can be taken to be highly saturated. The uniformity will be exploited in Chapter 12 to prove the existence of ultrapowers that are highly homogeneous (in addition to being highly saturated).

The results in this chapter are analogous to the Keisler-Shelah Theorem in ordinary model theory. (See [She71] and Chapter 6 in [CK90].) Moreover, our proof follows a similar line of argument, with adjustments appropriate to the handling of positive bounded formulas and their approximations.

The ideas and methods of model theory are being applied today in nearly all parts of mathematics. Here we concentrate on a framework for applications in functional analysis. Model theory has already provided several tools for the research analyst, of which the most important are: (a) the Banach space ultraproduct and nonstandard hull constructions in functional analysis; (b) spaces of (model-theoretic) types as used in the geometry of Banach spaces; and (c) Loeb measure spaces in stochastic analysis and its applications. In this paper we explain a systematic model theoretic framework within which these tools (especially (a) and (b)) are naturally situated. Our main intended audience consists of analysts who are familiar with the ultraproduct construction, and this perspective has strongly influenced our presentation of the material. We expect that many model theorists will also find something of interest in this subject. In particular, we indicate the initial steps of a program for introducing the key ideas and methods of model theory into functional analysis in a systematic and comprehensive way.

Applications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine [BDCK66] [Kri67] [DCK72] and nonstandard hulls by Luxemburg [Lux69b]. These two constructions Avere first used at about the same time and they are essentially the same; however, these initial steps led to largely independent lines of research that have still not been fully integrated.

1. Introduction

The game of Amazons was invented by Walter Zamkauskas. Two players with four playing pieces each compete on a 10 x 10 board. Figure 1 shows the initial position of the game. The pieces, called queens or amazons, move like chess queens. After each move an amazon shoots an arrow, which travels in the same way as a chess queen moves. The point where an arrow lands is burned off the playing board, reducing the effective playing area. Neither queens nor arrows can travel across a burned off square or another queen. The first player who cannot move with any queen loses.

Amazons endgames share many characteristics with Go endgames, but avoid the extra complexity of Go such as ko fights or the problem of determining the safety of stones and territories. Just like Go, Amazons endgames are being studied by combinatorial games researchers. Berlekamp and Snatzke have investigated play on sums of long narrow n x 2 strips containing one amazon of each player [1; 15]. Even though n x 2 areas have a simple structure, sum game play is surprisingly subtle, and full combinatorial game values become very complex.

1. Introduction

Cellular Automata Games have traditionally been 0-player games such as Conway's Life, or solitaire games played on a grid or digraph G = (V, E). (This includes undirected graphs, since every undirected edge {u, v} can be interpreted as the pair of directed edges (u,v) and (v,u).) Each cell or vertex of the graph can assume a finite number of possible states. The set of all states is the alphabet. We restrict attention to the binary alphabet {0,1}. A position is an assignment of states to all the vertices. There is a local transition rule from one position to another: pick a vertex u and fire it, i.e., complement it together with its neighborhood. The aim is to move from a given position (such as all 0s) to a target position (such as all 0s). In many of these games any order of the moves produces the same result, so the outcome depends on the set of moves, not on the sequence of moves. Two commercial manifestations are Lights Out manufactured by Tiger Electronics, and Merlin Magic Square by Parker Brothers (but Arthur-Merlin games are something else again). Quite a bit is known about such solitaire games.

1. Global Wins and Global Threats

Combinatorial game theory (CGT) applies the divide and conquer paradigm to game analysis and game tree search. We decompose a game into independent components (local games) and compute its value as the sum of all local games. The end of play in a sum of combinatorial games is determined by the normal termination rule: A player unable to move loses. Thus, in a sum of games, no single move or game can be decisive by itself. We investigate a class of games where a move in a local game may lead to an overall win in the sum of all local games. We call such a move globally winning.

Examples of globally winning moves are moves that capture a vital opponent piece such as checkmate, moves that promote a piece to a much more powerful one like promoting a king in checkers, or moves that “escape” in games where one side has to try to catch the other side's pieces like in the game Fox and Geese (Winning Ways [2], chapter 20). Figure 1 shows a Fox and Geese position where the fox escapes with his last move from e5 to d4 and obtains “an infinite number of free moves”. If this game were a component of a sum game S, the fox side would never lose in S.

1. Introduction

Hypercube tic-tac-toe is a two-person game played on an nk “board” (i.e. a /c-dimensional hypercube of side n). (The familiar 3 x 3 game has k = 2 and n = 3. Several editions of the 43 game, k = 3 and n = 4, are commercially available.) In all these games the two players take turns. Each player claims a single one of the nk cells with his/her symbol (traditionally O's and X's, or “noughts and crosses”, as the game is known in the UK), and the first player to complete a “path” of length n (in any straight line, including any type of diagonal) is the winner. If all nk cells are filled (with the two kinds of symbols) but no solid-symbol path has been completed, the game is declared a draw.

Since the first move cannot be a disadvantage, with best play the first player should never lose. Hence, in the ideal world, the first player seeks a win, while the second player tries to draw. For each given value of k, we believe there is a critical value nd of n below which the first player can force a win, while at or above this critical value, the second player can obtain a draw. This exact value of n is exceedingly difficult to determine as a function of k. (The larger the value of n for a given k, the easier it becomes for the second player to draw.)