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The traditional definition of a is that it is any closed subalgebra of the continuous functions on a compact Hausdorff space. For us—at least at the beginning—the relevant compact Hausdorff space is the circle T. Classically, an important function algebra has been —the functions continuous on and holomorphic on . [We call the Each such function can be identified with its restriction to the circle. And any such restriction has Fourier series with no coefficients of negative index. So this is clearly a subspace. It also follows by inspection that it is a subalgebra, and is closed.

Felix Klein’s lays out a blueprint for understanding a geometry by way of the mappings that preserve that geometry. This vision has become quite prevalent and powerful in modern approaches to the subject. Certainly Alexandre Grothendieck and Saunders Mac Lane carried this idea to new heights in their modern formulations of algebraic geometry and algebraic topology.

One of the most important things that we do in complex function theory is to construct holomorphic functions with specified properties. Given the way that we are prone to think, a natural way to effect this process is to perform some local construction and then to endeavor to extend the result to an entire domain The function theory of one complex variable is replete with methods for performing that “extension” process. Infinite products, analytic continuation, division problems, approximation theorems (Runge, Mergelyan), and the Cauchy—Riemann equations are just some of the devices that we have for taking a local construction and making it global.