Nonarchimedean Functional Analysis Peter Schneider Version: 25.10.2005 1 This book grew out of a course which I gave during the winter term 1997/98 at the Universit¨at M¨unster. The course covered the material which here is presented in the first three chapters. The fourth more advanced chapter was added to give the reader a rather complete tour through all the important aspects of the theory of locally convex vector spaces over nonarchimedean fields. There is one serious restriction, though, which seemed inevitable to me in the interest of a clear presentation. In its deeper aspects the theory depends very much on the field being spherically complete or not. To give a drastic example, if the field is not spherically complete then there exist nonzero locally convex vector spaces which do not have a single nonzero continuous linear form. Although much progress has been made to overcome this problem a really nice and complete theory which to a large extent is analogous to classical functional analysis can only exist over spherically complete fields. I therefore allowed myself to restrict to this case whenever a conceptual clarity resulted. Although I hope that this text will also be useful to the experts as a reference my own motivation for giving that course and writing this book was different. I had the reader in mind who wants to use locally convex vector spaces in the applications and needs a text to quickly grasp this theory. There are several areas, mostly in number theory like p-adic modular forms and deformations of Galois representations and in representation theory of p-adic reductive groups, in which one can observe an increasing interest in methods from nonarchimedean functional analysis. By the way, discretely valued fields like p-adic number fields as they occur in these applications are spherically complete. This is a textbook which is self-contained in the sense that it requires only some basic knowledge in linear algebra and point set topology. Everything presented is well known, nothing is new or original. Some of the material in the last chapter appears in print for the first time, though. In the references I have listed all the sources I have drawn upon. At the same time this list shows to the reader who the protagonists are in this area of mathematics. I certainly do not belong to this group. M¨unster, May 2001 Peter Schneider 2 List of contents Chap. I: Foundations §1 Nonarchimedean fields §2 Seminorms §3 Normed vector spaces §4 Locally convex vector spaces §5 Constructions and examples §6 Spaces of continuous linear maps §7 Completeness §8 Fr´echet spaces §9 The dual space Chap. II: The structure of Banach spaces §10 Structure theorems §11 Non-reflexivity Chap. III: Duality theory §12 c-compact and compactoid submodules §13 Polarity §14 Admissible topologies §15 Reflexivity §16 Compact limits Chap. IV: Nuclear maps and spaces §17 Topological tensor products §18 Completely continuous maps §19 Nuclear spaces §20 Nuclear maps §21 Traces §22 Fredholm theory References Index, Notations 3 Chap. I: Foundations In this chapter we introduce the basic notions and constructions of nonar- chimedean functional analysis. We begin in §1 with a brief but self-contained review of nonarchimedean fields. The main objective of functional analysis is the investigation of a certain class of topological vector spaces over a fixed nonar- chimedean field K. This is the class of locally convex vector spaces. The more traditional analytic point of view characterizes locally convex topologies as those vector space topologies which can be defined by a family of (nonarchimedean) seminorms. But the presence of the ring of integers o inside the field K allows for an equivalent algebraic point of view. A locally convex topology on a K-vector space V is a vector space topology defined by a class of o-submodules of V which are required to generate V as a vector space. In §§2 and 4 we thoroughly discuss these two concepts and their equivalence. Throughout the book we usually will present the theory from both angles. But sometimes there will be a certain bias towards the algebraic point of view. The most basic methods to actually construct locally convex vector spaces along with concrete examples are treated in §§3 and 5. In §6 we explain how the notion of a bounded subset leads to a systematic method to equip the vector space of continuous linear maps between two given locally convex vector spaces with a natural class of locally convex topologies. The two most important ones among them are the weak and the strong topology. The important concepts of completeness and quasi-completeness are discussed in §7. The construction of the completion of a locally convex vector space is one of the places where we find an algebraic treatment preferable since conceptually simpler. Banach spaces as already introduced in §3 are complete. They are included in the very important class of Fr´echet spaces. These are the complete locally convex vector spaces whose topology is metrizable. Their importance partly derives from the validity of the closed graph and open mapping theorems for linear maps between Fr´echet spaces. These basic results are established in §8 using Baire category theory. In the final §9 of this chapter we begin the investigation of the continuous linear dual of a locally convex vector space. Provided the field K is spherically complete we establish the Hahn-Banach theorem about the existence of continuous linear forms. This is then applied to obtain the first properties of the duality maps into the various forms of the linear bidual. In this section we encounter for the first time the phenomenon in nonarchimedean functional analysis that crucial aspects of the theory depend on special properties of the nonarchimedean field K. The ultimate reason for this difficulty is that K need not to be locally compact. A satisfactory substitute for compact subsets in locally convex K- vector spaces only exists if the field K is spherically complete. This will be discussed systematically in §12 of the third chapter. §1 Nonarchimedean fields 4 Let K be a field. A nonarchimedean absolute value on K is a function | | : K −→ IR such that, for any a,b ∈ K we have (i) |a| ≥ 0, (ii) |a| = 0 if and only if a = 0, (iii) |ab| = |a|·|b|, (iv) |a + b| ≤ max(|a|, |b|). The condition (iv) is called the strict triangle inequality. Because of (iii) the × × map | | : K −→ IR+ is a homomorphism of groups. In particular we have |1| = | − 1| = 1. We always will assume in addition that | | is non-trivial, i.e., that (v) there is an a0 ∈ K such that |a0| 6=0, 1. It follows immediately that |n · 1| ≤ 1 for any n ∈ ZZ. Moreover, if |a| 6= |b| for some a,b ∈ K then the strict triangle inequality actually can be sharpened into the equality |a + b| = max(|a|, |b|) . To see this we may assume that |a| < |b|. Then |a| < |b| = |b + a − a| ≤ max(|b+a|, |a|), hence |a| < |a+b| and therefore |b|≤|a+b| ≤ max(|a|, |b|) = |b|. Via the distance function d(a,b) := |b − a| the set K is a metric and hence topological space. The subsets Bǫ(a) := {b ∈ K : |b − a| ≤ ǫ} for any a ∈ K and any real number ǫ > 0 are called closed balls or simply balls in K. They form a fundamental system of neighbourhoods of a in the metric space K. Likewise the open balls − Bǫ (a) := {b ∈ K : |b − a| < ǫ} form a fundamental system of neighbourhoods of a in K. As we will see below − Bǫ(a) and Bǫ (a) are both open and closed subsets of K. Talking about open and closed balls therefore does not refer to a topological distinction but only to the nature of the inequality sign in the definition. We point out the following two simple facts. 1) If | |∞ denotes the usual archimedean absolute value on IR then, for any b ∈ − Bǫ (a), we have ||b|−|a||∞ = ||(b−a)+a|−|a||∞ ≤|max(|b−a|, |a|)−|a||∞ < ǫ. This means that the absolute value | | : K −→ IR is a continuous function. − − − 2) For b0 ∈ Bǫ (a0) and b1 ∈ Bǫ (a1) we have b0 + b1 ∈ Bǫ (a0 + a1) and b b ∈ B− (a a ). The latter follows from b b − a a = (b − 0 1 ǫ·max(ǫ,|a0|,|a1|) 0 1 0 1 0 1 0 a0)(b1 −a1)+(b0 −a0)a1 +a0(b1 −a1). This says that addition + : K ×K −→ K and multiplication · : K × K −→ K are continuous maps. 5 Lemma 1.1: i. Bǫ(a) is open and closed in K; ′ ′ ii. if Bǫ(a) ∩ Bǫ(a ) 6= ∅ then Bǫ(a) = Bǫ(a ); iii. If B and B′ are any two balls in K with B ∩ B′ 6= ∅ then either B ⊆ B′ or B′ ⊆ B; iv. K is totally disconnected. Proof: The assertions i. and ii. are immediate consequences of the strict triangle inequality. The assertion iii. follows from ii. To see iv. let M ⊆ K be a nonempty connected subset. Pick a point a ∈ M. By i. the intersection M ∩ Bǫ(a) is open and closed in M. It follows that M is contained in any ball around a and therefore must be equal to {a}. Clearly the assertions i.-iii. hold similarly for open balls. The assertion ii. says that any point of a (open) ball can serve as its midpoint. On the other hand the real number ǫ is not uniquely determined by the set Bǫ(a) and therefore cannot be considered as the ”radius” of this ball.