By now, even a skeptical reader should be thoroughly sold on the utility of embeddings into cubes. But the sales pitch is far from over! We next pursue the consequences of a result stated earlier: If X is completely regular, then Cb(X) completely determines the topology on X. In brief, to know Cb(X) is to know X. Just how far can this idea be pushed? If Cb(X) and Cb(Y) are isomorphic (as Banach spaces, as lattices, or as rings), must X and Y be homeomorphic? Which topological properties of X can be attributed to structural properties of Cb(X) (and conversely)?

These questions were the starting place for Marshall Stone's 1937 landmark paper, “Applications of the Theory of Boolean Rings to General Topology” [140]. It's in this paper that Stone gave his account of the Stone– Weierstrass theorem, the Banach–Stone theorem, and the Stone– Čech compactification. (These few are actually tough to find among the dozens of results in this mammoth 106-page work.) A signal passage from his introduction may be paraphrased as follows: “We obtain a reasonably complete algebraic insight into the structure of Cb(X) and its correlation with the structure of the underlying topological space.” Stone's work proved to be a gold mine – the digging continued for years! – and its influence on algebra, analysis, and topology alike can be seen in virtually every modern textbook.

Independently, but later that same year (1937), Eduard Čech [24] gave another proof of the existence of the compactification but, strangely, credits a 1929 paper of Tychonoff for the result (see Shields [136] for more on this story).

As pointed out earlier, the spaces Lp and lp exhaust the “isomorphic types” of Lp(μ) spaces. Thus, to better understand the isomorphic structure of Lp(μ) spaces, we might ask, as Banach did:

For p ≠ r, can lr or Lr be isomorphically embedded into Lp?

We knowquite a bit about this problem.We knowthat the answer is always yes for r = 2, and, in case 2 < p < ∞, the Kadec–Pelczyński theorem (Corollary 9.7) tells us that r = 2 is the only possibility. In this chapter we'll prove the following statement:

If p and r live on opposite sides of 2, there can be no isomorphism from Lr or lr into Lp.

This leaves open only the cases 1 = r < p < 2 and 1 ≤ p < r < 2. The first case can also be eliminated, as we'll see, but not the second.

Unconditional Convergence

We next introduce the notion of unconditional convergence of series. What follows are several plausible definitions.

We say that a series Σn xn in a Banach space X is

1. (a) unordered convergent if Σn xπ(n) converges for every permutation (one–to–one, onto map) π : ℕ → ℕ;

2. (b) subseries convergent if Σk xnk converges for every subsequence (xnk) of (xn);

3. (c) “random signs” convergent if Σn εn xn converges for any choice of signs εn = ±1;

4. (d) bounded multiplier convergent if Σn an xn converges whenever |an| ≤ 1.

The spaces c0, L1, L2, and L∞; play very special roles in Banach space theory. You're already familiar with the space l and its unique position as the sole Hilbert space in the family of lp spaces. We won't have much to say about l2 here. And by now you will have noticed that the space l∞ doesn't quite fit the pattern that we've established for the other lp spaces. for one, it's not separable and so doesn't have a basis. Nevertheless, we will be able to say a few meaningful things about l∞. The spaces c0 and l1, on the other hand, play starring roles when it comes to questions involving bases in Banach spaces and in the whole isomorphic theory of Banach spaces for that matter. Unfortunately, we can't hope to even scratch the surface here. But at least a few interesting results are within our reach.

Throughout, (en) denotes the standard basis in c0 or l1, and denotes the associated sequence of coefficient functionals. As usual, (en) and are really the same; we just consider them as elements of different spaces.

True Stories Aboutl1

We begin with a “universal” property of l1 due to Banach and Mazur [8].

Theorem 6.1.Every separable Banach space is a quotient of l1.

Proof. Let X be a separable Banach space, and write = {x : ∥ x ∥ < 1} to denote the open unit ball in X.

To begin, recall that a Banach space is a complete normed linear space. That is, a Banach space is a normed vector space (X, ∥ · ∥) that is a complete metric space under the induced metric d(x, y) = ∥ x – y ∥. Unless otherwise specified, we'll assume that all vector spaces are over ℝ, although, from time to time, we will have occasion to consider vector spaces over ℂ.

What follows is a list of the classical Banach spaces. Roughly translated, this means the spaces known to Banach. Once we have these examples out in the open, we'll have plenty of time to fill in any unexplained terminology. For now, just let the words wash over you.

The Sequence Spaceslpandc0

Arguably the first infinite-dimensional Banach spaces to be studied were the sequence spaces lp and c0. To consolidate notation, we first define the vector space s of all real sequences x = (xn) and then define various subspaces of s.

For each 1 ≤ p < ∞, we define

and take lp to be the collection of those x ∈ s for which ∥ x ∥p < ∞. The inequalities of Hölder and Minkowski show that lp is a normed space; from there it's not hard to see that lp is actually a Banach space.

The space lp is defined in exactly the same way for 0 < p < 1 but, in this case, ∥ · ∥p defines a complete quasi-norm. That is, the triangle inequality now holds with an extra constant; specifically, ∥ x + y ∥p ≤ 21/p(∥ x∥p + ∥ y ∥y).

We begin with a brief summary of important facts from functional analysis – some with proofs, some without. Throughout, X, Y, and so on, are normed linear spaces over ℝ. If there is no danger of confusion, we will use ∥ · ∥ to denote the norm in any given normed space; if two or more spaces enter into the discussion, we will use ∥ · ∥X, and so forth, to further identify the norm in question.

Continuous Linear Operators

Given a linear map T : X → Y, recall that the following are equivalent:

1. (i) T is continuous at 0 ∈ X.

2. (ii) T is continuous.

3. (iii) T is uniformly continuous.

4. (iv) T is Lipschitz; that is, there exists a constant C < ∞ such that ∥ T x – T y ∥Y ≤ C∥ x – y ∥X for all x, y ∈ X.

5. (v) T is bounded; that is, there exists a constant C < ∞ such that ∥T x∥Y ≤ C∥x∥X for all x ∈ X.

If a linear map T : X → Y is bounded, then there is, in fact, a smallest constant C satisfying ∥T x∥Y ≤ C∥x∥X for all x ∈ X. Indeed, the constant

called the norm of T, works; that is, it satisfies ∥T x∥Y ≤ ∥T ∥ ∥x∥X and it's the smallest constant to do so. Further, it's not hard to see that (2.1) actually defines a norm on the space B(X, Y) of all bounded, continuous, linear maps T : X → Y.

In this chapter we present Garling's proof [53] of the Riesz representation theorem for the dual of C(K), K compact Hausdorff. This theorem goes by a variety of names: The Riesz–Markov theorem, the Riesz–Kakutani theorem, and others. The version that we'll prove states:

Theorem 16.1.Let K be a compact Hausdorff space, and let T be a positive linear functional on C(K). Then there exists a unique positive Baire measure μ on K such that T (f) = ∫K f d μ for every f ∈ C(K).

As we pointed out in the last chapter, our approach will be to first prove the theorem for l∞ spaces. To this end, we will need to know a bit more about the Stone–Čech compactification of a discrete space and a bit more measure theory. First the topology.

The Stone–Čech Compactification of a Discrete Space

A topological space is said to be extremally disconnected, or Stonean, if the closure of every open set is again open. Obviously, discrete spaces are extremally disconnected. Less mundane examples can be manufactured from this starting point:

Lemma 16.2.If D is a discrete space, then βD is extremally disconnected.

Proof. Let U be open in βD, and let A = U ∩ D. Then A is dense in U since U is open, and so clβD A = clβDU. Now we just check that clβDA is also open. The characteristic function χA : D → {0, 1} (a continuous function on D!) extends continuously to some f : βD → {0, 1}. Thus, by continuity, clβDA = f-1({1}) is open.

These are notes for a graduate topics course offered on several occasions to a rather diverse group of doctoral students at Bowling Green State University. An earlier version of these notes was available through my Web pages for some time and, judging from the e-mail I've received, has found its way into the hands of more than a few readers around the world. Offering them in their current form seemed like the natural thing to do.

Although my primary purpose for the course was to train one or two students to begin doing research in Banach space theory, I felt obliged to present the material as a series of compartmentalized topics, at least some of which might appeal to the nonspecialist. I managed to cover enough topics to suit my purposes and, in the end, assembled a reasonable survey of at least the rudimentary tricks of the trade.

As a prerequisite, the students all had a two-semester course in real analysis that included abstract measure theory along with an introduction to functional analysis. While abstract measure theory is only truly needed in the final chapter, elementary facts from functional analysis, such as the Hahn–Banach theorem, the Open Mapping theorem, and so on, are needed throughout. Chapter 2, “Preliminaries,” offers a brief summary of several key ideas from functional analysis, but it is far from self-contained. This chapter also features a large set of exercises I used as the basis for additional review, when necessary. A modest background in topology is also helpful but, because many of my students needed review here, I included a brief appendix containing most of the essential facts.