We have defined a vector space (§5.7) where each vector in the space is a list of 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 multiplied by scalars is called a linear combination. Vector spaces are closed under linear combinations. That is, the linear combination is also in the space ( ), for any positive integer , any , and any . More generally, vector spaces can be defined over any field of scalars and any set of vectors in which vector addition forms an abelian group [58, pp. 282-291]. In this book, we only need and .
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 (``contains its limit points''), and a complete vector space with a norm defined on it is called a Banach space. Our vector space 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  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.