Andrew Browder

Point derivations and analytic structure in the spectrum of a Banach algebra

Abstract Let A be a commutative complex Banach algebra with unit. Via the Gelfand map, ƒ → ƒ , A may be represented as an algebra of continuous functions on Spec A, the spectrum (maximal ideal space) of A. It is shown that if the space of point derivations on A at θ ϵ Spec A has finite dimension n, then there exists an analytic variety V in a polycylinder in Cn and a homeomorphism τ of V onto a neighborhood of θ (in the metric topology which Spec A inherits from A ∗ ), such that ƒ ○ τ is analytic on V for every ƒ in A, This generalizes a theorem of Gleason.

Integration on Manifolds

In this chapter, we define the integral of a k-form over a compact oriented k-manifold, and prove the important generalized Stokes’ theorem, which can be regarded as a far-reaching generalization of the fundamental theorem of calculus. We also define the integral of a function over a (not necessarily oriented) manifold, and describe the integral of a form in terms of the integral of a function. The classical theorems of vector analysis (Green’s theorem, divergence theorem, Stokes’ theorem) appear as special cases of the general Stokes’ theorem. Applications are made to topology (the Brouwer fixed point theorem) and to the study of harmonic functions (the mean value property, the maximum principle, Liouville’s theorem, and the Dirichlet principle).

Continuous Functions on Intervals

In this chapter, we begin the study of continuous functions with the special case of real-valued functions defined on an interval in R. The concepts we develop here will be reexamined in a more general setting in later chapters.

The Riemann Integral

In this chapter we give an exposition of the definite integral of a real-valued function defined on a closed bounded interval. We assume familiarity with this concept from a previous study of calculus, but want to develop the theory in a more precise way than is typical for calculus courses, and also take a closer look at what kind of functions can be integrated. The integral to be defined and studied here is now widely known as the Riemann integral; in a later chapter we will study the more general Lebesgue integral.

Sequences and Series

In the first chapter, we defined a sequence in X to be a mapping from N to X. Let us broaden this definition slightly, and allow the mapping to have a domain of the form \(\left\{ {n \in {\rm Z}:m\underline m} \right\}\), for some m ∈ Z (usually, but not always, m = 0 or m = 1). The most common notation is to write n→ xn instead of n→ x(n). If the domain of the sequence is the finite set \(\left\{ {m,m + 1, \ldots ,p} \right\}\), we write the sequence as \(\left( {x_n } \right)_{n = m}^p\), and speak of a finite sequence (though we emphasize that the sequence should be distinguished from the set \(\left. {\left\{ {x_n :m\underline m} } \right\}\), we write it as \((x_n )_{n = m}^\infty\), and speak of an infinite sequence. Note that the corresponding set of values \(\left\{ {x_n :n\underline > m} \right\}\) may be finite. When the domain of the sequence is understood from the context, or is not relevant to the discussion, we write simply (xn). In this chapter, we shall be concerned with infinite sequences in R.