ULTRAPRODUCTS IN ANALYSIS JUNG JIN LEE Abstract. Basic concepts of ultraproduct and some applications in Analysis, mainly in Banach spaces theory, will be discussed. It appears that the concept of ultraproduct, originated as a fundamental method of a model theory, is widely used as an important tool in analysis. When one studies local properties of a Banach space, for example, these constructions turned out to be useful as we will see later. In this writing we are invited to look at some basic ideas of these applications. 1. Ultrafilter and Ultralimit Let us start with the de¯nition of ¯lters on a given index set. De¯nition 1.1. A ¯lter F on a given index set I is a collection of nonempty subsets of I such that (1) A; B 2 F =) A \ B 2 F , and (2) A 2 F ;A ½ C =) C 2 F . Proposition 1.2. Each ¯lter on a given index set I is dominated by a maximal ¯lter. Proof. Immediate consequence of Zorn's lemma. ¤ De¯nition 1.3. A ¯lter which is maximal is called an ultra¯lter. We have following important characterization of an ultra¯lter. Proposition 1.4. Let F be a ¯lter on I. Then F is an ultra¯lter if and only if for any subset Y ½ I, we have either Y 2 F or Y c 2 F . Proof. (=)) Since I 2 F , suppose ; 6= Y2 = F . We have to show that Y c 2 F . De¯ne G = fZ ½ I : 9A 2 F such that A \ Y c ½ Zg. Since Y2 = F , for any A 2 F , A \ Y c 6= ;. Therefore, G is a collection of nonempty subsets of I. Moreover, c c (1) If Z1;Z2 2 G , then we can ¯nd A1;A2 2 F such that A1 \ Y ½ Z1, A2 \ Y ½ Z2. c Now A1 \ A2 2 F and A1 \ A2 \ Y ½ Z1 \ Z2, i.e., Z1 \ Z2 2 G . (2) If Z1 ½ Z2 and Z1 2 G , then it's clear that Z2 2 G . So G is a ¯lter. Since A ⊃ A \ Y c for any A 2 F , we have F ½ G , and hence F = G by maximality of F . Noting that Y c 2 G = F , we are done. ((=) Suppose G is a ¯lter such that F ½ G . Suppose to the contrary that 9Y 2 G n F . From our assumption, Y c 2 F and hence Y c 2 G . Then Y \Y c 2 G , which is impossible. ¤ Date: August 3, 2004. This document can be obtained at http://www.math.uiuc.edu/»junglee2/misc/ultraproduct.pdf. 1 2 JUNG JIN LEE Remark An ultra¯lter U is called trivial (or non-free, principal, or ¯xed) if it is generated by a single element i0 2 I, that means, I0 2 U if and only if i0 2 I0. An ultra¯lter which is not trivial is called free. Another notion is of importance for several constructions: an ultra¯lter U is called countably incomplete if there is a sequence of elements of U satisfying \1 I1 ¶ I2 ¶ ¢ ¢ ¢ ; Ik = ;: k=1 Some authors just say ultra¯lters when mentioning countably incomplete ultra¯lters, because it's the most interesting case. If I is ¯nite, every ultra¯lter is ¯xed. If I is in¯nite, then there is a free ultra¯lter on I. See [2]. De¯nition 1.5. Let F be a ¯lter on I. A set of real numbers (xi)i2I is said to converge to x 2 R with respect to F if for all ² > 0, fi 2 I : jxi ¡ xj < ²g 2 F . In this case, x is called the ultralimit of (xi)i2I , denoted x = limU xi. Remark If such x exists in the previous de¯nition, it is unique, since F consists of nonempty sets. So this de¯nition makes sense. The following is one of the important features of ultralimits. Theorem 1.6. Let U be an ultra¯lter on I and (xi)i2I be a set of bounded real numbers, then (xi)i2I must converge. + ¡ Proof. Suppose jxij · M, 8i 2 I. Let S0 = fi 2 I : xi ¸ 0g, S0 = fi 2 I : xi · 0g. ¡ + c + Noting S0 ⊃ (S0 ) , we know that at least one of these should be in U ; say S0 and set + M ¡ M T0 = [0;M]. De¯ne S1 = fi 2 I : xi ¸ 2 g, S1 = fi 2 I : xi · 2 g. Just like before, we may ¡ M + M assume that, say, S1 2 U ; set T1 = [0; 2 ]. In a like fashion, de¯ne S2 = fi 2 I : xi ¸ 4 g, ¡ M + M M S2 = fi 2 I : xi · 4 g. Assume S2 2 U and set T2 = [ 4 ; 2 ]. Iterate this process. By the T1 Principle of Nested Intervals, we get a point x 2 R such that k=0 Ik = fxg. We claim that lim x = x. Fix ² > 0 and consider the set fi 2 I : jx ¡ xj < ²g. Choose N so that M < ². U i T i 2N M N ~ ~ Since the length of TN = 2N , we have k=0 Sk ½ fi 2 I : jxi ¡ xj < ²g, where Sk is one of + ¡ Sk or Sk whichever is in U . ¤ Above theorem is true in more general setting; in fact, we have the following result. Theorem 1.7. Let K be a compact Hausdor® space. Then for each family (xi)i2I with xi 2 K the limit lim xi = x U exists in K. This means, there is a unique point x 2 K such that, for each neighborhood V of x, the set fi 2 I : xi 2 V g belongs to U . Proof. Refer to [2]. ¤ Remark Actually, a basic fact from general topology is that X is a compact Hausdor® space if and only if for every indexed family (xi)i2I in X and every ultra¯lter U on I the ultralimit of (xi)i2I exists and is unique. See [4] Usual rules of limits also work for ultralimits. More precisely, we have following proposition. We omit its proof since it is just a simple check of de¯nition. Proposition 1.8. For ultralimits, we have that ULTRAPRODUCTS IN ANALYSIS 3 (1) limU jaxij = jaj ¢ limU jxij. (2) limU jxi + yij · limU jxij + limU jyij. (3) If xi · yi for any i, then limU xi · limU yi. (4) limU (xi + yi) = limU xi + limU yi. (5) limU C = C for any C constant. (6) limU xi ¢ yi = limU xi ¢ limU yi. Remark (3)+(4) gives a proof of (2). We close this section with a proposition about the product of ultra¯lters. Proposition 1.9. Let U ; V be ultra¯lters on index sets I;J, respectively. Then U £ V is an ultra¯lter on I £ J where U £ V is de¯ned by: X 2 U £ V if and only if fj : fi :(i; j) 2 Xg 2 U g 2 V : Proof. First we show that U £V is a ¯lter on I £J. Note that U £V consists of nonempty sets. (1) Suppose A; B 2 U £V . We want to show that fj : fi :(i; j) 2 A\Bg 2 U g 2 V . Let A~ = fj : fi :(i; j) 2 Ag 2 U g and B~ = fj : fi :(i; j) 2 Bg 2 U g. Since A;~ B~ 2 V , ~ ~ ~ ~ A \ B 2 V . Choose j0 2 A \ B and let Aj0 = fi :(i; j0) 2 Ag, Bj0 = fi :(i; j0) 2 Bg. Now Aj0 ;Bj0 2 U so Aj0 \ Bj0 = fi :(i; j0) 2 A \ Bg 2 U . This shows that A~ \ B~ ½ fj : fi :(i; j) 2 A \ Bg 2 U g and hence fj : fi :(i; j) 2 A \ Bg 2 U g 2 V . (2) Suppose A 2 U £ V and A ½ C. Choose j0 2 fj : fi :(i; j) 2 Ag 2 U g. Since A ½ C, j0 2 fj : fi :(i; j) 2 Cg 2 U g. Thus fj : fi :(i; j) 2 Ag 2 U g(2 V ) ½ fj : fi :(i; j) 2 Cg 2 U g, which shows C 2 V . To show that U £ V is maximal, pick any X ½ I £ J and suppose X2 = U £ V . By maximality of V , fj : fi :(i; j) 2 Xg 2= U g 2 V . Choose j0 2 fj : fi :(i; j) 2 Xg 2= U g c and consider the set fi :(i; j0) 2 Xg. By maximality of U , we have fi :(i; j0) 2 X g 2 U , which shows that fj : fi :(i; j) 2 Xg 2= U g(2 V ) ½ fj : fi :(i; j) 2 Xcg 2 U g. Thus fj : fi :(i; j) 2 Xcg 2 U g 2 V . ¤ 2. Ultraproducts of Banach Spaces Let (Ei)i2I be a family of Banach spaces. Consider the space `1(I;Ei) of families (xi)i2I with xi 2 Ei(i 2 I) and k(xi)k1 = sup kxik < 1: i2I Then `1(I;Ei) is a Banach space, which is too big in some sense. Let NU be the subset of all those families (xi)i2I 2 `1(I;Ei) with limU kxik = 0. One can easily verify that NU is a closed linear subspace of `1(I;Ei). So we are ready to give De¯nition 2.1. The ultraproduct (Ei)U of the family of Banach spaces (Ei)i2I with respect to the ultra¯lter U is the quotient space `1(I;Ei)=NU ; equipped with the canonical quotient norm. 4 JUNG JIN LEE Q Remark Often the ultraproduct (Ei)U is also denoted by i2I Ei=U .