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 addition (§5.3) and scalar multiplication (§5.5). 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.