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Summary and Related Mathematical Topics

We have defined a vector space $ \mathbb{C}^N$5.7) where each vector in the space is a list of $ N$ complex numbers. There are two operations we can perform on vectors: vector addition5.3) and scalar multiplication5.5). The sum of two or more vectors $ \underline{x}_i\in\mathbb{C}^N$ multiplied by scalars $ \alpha_i\in\mathbb{C}$ is called a linear combination. Vector spaces are closed under linear combinations. That is, the linear combination $ \underline{y}= \alpha_1\,\underline{x}_1
+ \alpha_2\,\underline{x}_2 + \cdots + \alpha_M\,\underline{x}_M$ is also in the space $ \mathbb{C}^N$ for any positive integer $ M$ , any $ \alpha_i\in\mathbb{C}$ , and any $ \underline{x}_i\in\mathbb{C}^N$ . More generally, vector spaces can be defined over any field $ F$ of scalars and any set of vectors $ V$ in which vector addition forms an abelian group [58, pp. 282-291]. In this book, we only need $ F=\mathbb{C}$ and $ V=\mathbb{C}^N$ .

We have introduced the usual Eucidean norm to define vector length. When every Cauchy sequence is convergent to a point in the space, the space is said to be complete (``it contains its limit points''), and a complete vector space with a norm defined on it is called a Banach space. Our vector space $ \mathbb{C}^N$ with the Euclidean norm defined on it qualifies as a Banach space. We will next define an inner product on the space, which will give us a Hilbert space. Formal proofs of such classifications are beyond the scope of this book, but [58] and Web searches on the above terms can spur further mathematical study. It is also useful to know the names (especially ``Hilbert space''), as they are often used.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University