Summary and Related Mathematical Topics

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.

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