# A Guide to Functional Analysis

Steven G. Krantz
Volume: 49
Edition: 1
Pages: 150
https://www.jstor.org/stable/10.4169/j.ctt6wpwrd

1. Front Matter
(pp. I-VIII)
(pp. IX-X)
3. Preface
(pp. XI-XII)
Steven G. Krantz
4. CHAPTER 1 Fundamentals
(pp. 1-26)

The mathematical analysts of the nineteenth century (Cauchy, Riemann, Weierstrass, and others) contented themselves with studying one function at a time. As a sterling instance, the Weierstrass nowhere differentiable function is a world-changing example of the real function theory of “ one function at a time.” Some of Riemann’s examples in Fourier analysis give other instances. This was the world view 150 years ago. To be sure, Cauchy and others considered sequences and series of functions, but the end goal was to consider the single limit function.

A major paradigm shift took place, however, in the early twentieth century.For then...

5. CHAPTER 2 Ode to the Dual Space
(pp. 27-32)

IfXis a normed linear space, then the collection of bounded linear functionals onXis called itsdual spaceand is denotedX*. Given a Banach spaceX, we frequently want to calculateX*. Knowledge ofX* can give us a good deal of information aboutXitself.

We note that, whenXis a Banach space, thenX* will also be a Banach space. If$\alpha \in X*$then

$||\alpha |X* \equiv \sup |\alpha (x)|$.

$||x|| \le 1$

When$\alpha \in X*$and$x \in X$then we sometimes write$\alpha (x)$for the action of α onxand sometimes write$\left\langle {\alpha ,} \right.\left. x \right\rangle$or$\left\langle {x,} \right.\left. \alpha \right\rangle$(in order to emphasize the dual...

6. CHAPTER 3 Hilbert Space
(pp. 33-44)

If we place some additional structure on a Banach space, then a richer theory results. That is what we now do.

The key idea is to equip the Banach space with an inner product. That gives us notions of orthogonality and projection, and the result is a beautiful and coherent theory.

Definition. A complex vector spaceHis called aninner product spaceif, to each ordered pair of vectors$x,y \in H$, there is associated a complex number$\left\langle {x,} \right.\left. y \right\rangle$. This number is called theinner productorscalar productofxandy. In the language of algebra, we may think...

7. CHAPTER 4 The Algebra of Operators
(pp. 45-58)

Before we begin the subject proper of this chapter, we need briefly to discuss some topological issues. In particular, we must treat some topologies onXandX*.

Definition. LetXbe a Banach space andX* its dual. We say that a sequence$\left\{ {{x_j}} \right\}$inXconverges to$x \in X$in theweak topologyif$\varphi ({x_j}) \to \varphi (x)$for every$\varphi \in X*$.

Definition. LetXbe a Banach space andX* its dual. We say that a sequence$\left\{ {{\varphi _j}} \right\}$inX* converges to$\varphi \in X*$in theweak-*topologyif${\varphi _j}(x) \to \varphi (x)$for every$x \in X$.

We can see that, in a certain sense, weak convergence...

8. CHAPTER 5 Banach Algebra Basics
(pp. 59-74)

The idea of a Banach algebra was conceived by I. M. Gelfand in his Ph.D. thesis of 1936. It is a beautiful blend of functional analysis and classical hard analysis. Particularly striking is how quickly and easily it leads to profound and elegant results. We shall present some of these in the present chapter.

Analgebrais a collection of objects equipped with binary operations of addition and multiplication, and also with a notion of scalar multiplication. For example, the collection of polynomials$p(x)$of one variable forms an algebra. This is clearly an algebraic idea. Gelfand’s key insight was...

9. CHAPTER 6 Topological Vector Spaces
(pp. 75-80)

The most basic mathematical structure for functional analysis is that of a Banach space. We have learned that the richer structure of Hilbert space can lead to greater depth and insight. Certainly the most elementary structure—more primitive than either Banach space or Hilbert space, but which still yields useful results—is that of topological vector space. We shall discuss the basic ideas here.

Convexity was implicit in much of what we did with Banach and Hilbert spaces. In the current presentation, convexity will play a more explicit role. We say that a setEin a linear space is...

10. CHAPTER 7 Distributions
(pp. 81-88)

The idea of generalized function has roots in the nineteenth century. Even more than 100 years ago, mathematicians wanted a way to say that a function satisfies a differential equation in some “weak” sense. The impetus was to develop a notion of function that is not conceived “point by point” as we usually do. This set of ideas developed further traction in the twentieth century, especially because of various questions in harmonic and functional analysis. It was in 1950 that Laurent Schwartz wrote his definitive book [SCH] enunciating the theory of distributions (or generalized functions).

11. CHAPTER 8 Spectral Theory
(pp. 89-98)

Certainly one of the premier theorems in all of functional analysis is the spectral theorem. It says that, on Hilbert spaceH, any reasonable bounded linear operator can be represented as multiplication by an${L^\infty }$function (acting on). Much of the modern theory, especially the theory of norm

al operators, depends critically on the spectral theorem.

There are many versions of the spectral theorem, both for bounded and for unbounded operators. Here, in the spirit of simplicity, we concentrate on a basic version for bounded operators.

Some preliminary, background terminology is this:

Definition. An operator$T \in B(H)$is said to...

12. CHAPTER 9 Convexity
(pp. 99-104)

In this chapter we treat the classical notion of convexity. Indeed, convexity is an old idea. It occurs in some of Archimedes’s treatments of the concept of arc length. But the idea of convexity was not actually formalized until the treatise [BOF] appeared in 1934. Since then it has been studied intensely, both in the classical Euclidean setting (see [KRA4]) and in the more general setting of infinite dimensions (see, for instance, [BAP], [SIM]).

Definition. LetXbe a topological vector space, and let$E \subseteq X$. We say thatEisconvexif, whenever$x,y \in X$, then

$(1 - t)x + ty \in E$for all$0 \le t \le 1$.

Example....

13. CHAPTER 10 Fixed-Point Theorems
(pp. 105-114)

Of course the granddaddy of all fixed-point theorems is that of L. E. J. Brouwer, proved in the early twentieth century. It says this:

Theorem 10.1.Let F be a continuous mapping of the closed unit ball in${R^N}$to itself. Then there is a point P in the closed unit ball such that F(P) = P.

Popular expositors have fun explaining the two-dimensional version of this theorem in terms of stirring a bowl of soup with grated cheese on top.

Such frivolity tends to disguise the fact that the fixed-point theorem is a profound result of mathematical analysis, and...

14. Table of Notation
(pp. 115-118)
15. Glossary
(pp. 119-128)
16. Bibliography
(pp. 129-132)
17. Index
(pp. 133-136)