Square Summable Power Series

Louis de Branges, James Rovnyak

• Publisher: Dover Publications
• Year: 2014
• ISBN:
  9780486801360
• Topic: Mathematics
• Subtopic: Complex Analysis
• Index: Mathematics

This text for advanced undergraduate and graduate students introduces Hilbert space and analytic function theory, which is centered around the invariant subspace concept. The book's principal feature is the extensive use of formal power series methods to obtain and sometimes reformulate results of analytic function theory.
The presentation is elementary in that it requires little previous knowledge of analysis, but it is designed to lead students to an advanced level of performance. This is achieved chiefly through the use of problems, many of which were proposed by former students. The book's tried-and-true approach was developed from the authors' lecture notes on courses taught at Lafayette College, Bryn Mawr College, and Purdue University.

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Theory of Formal Power Series

Any work with Hilbert space or analytic function theory requires an understanding of the complex number system. The complex numbers are pairs a + ib of real numbers a and b. By definition

The addition and multiplication of complex numbers have all the familiar properties of real numbers, known as the field postulates.¹ In addition to these properties, the complex numbers have a new property, called conjugation. The conjugate of a + ib is a – ib, and we write a – ib = (a + ib)^–. Note that the product of a complex number and its conjugate is positive,

except when a + ib = 0. The absolute value of a + ib is

where, as usual, the square root sign refers to the positive choice of root. The distance between complex numbers a + ib and c + id is

which is the ordinary Euclidean distance when a complex number is thought of as a point in the plane.

A sequence (a_n + ib_n) of complex numbers is said to converge to a complex number a + ib if for any given ∊ > 0, no matter how small, there is some number N, depending on ∊, such that

whenever

. A sequence (a_n + ib_n) of complex numbers is said to be Cauchy...