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 (``it 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 [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.