Signals and Hilbert Spaces - Signal Processing for Communications - page 41

N and length- N signal is read-
ily apparent, the question now hinges on how we are to proceed in order to
generalize the above concepts to the class of infinite sequences. Intuitively,
for instance, we can let N grow to infinity and obtain ∞ as the Euclidean
space for infinite sequences; in this case, however, much care must be ex-
ercised with expressions such as (3.1) and (3.3) which can diverge for se-
quences as simple as x

While the analogy between vectors in

l g r , y i d . , © , L s

[ n ]= 1forall n . In fact, the proper generalization of

N to an infinite number of dimensions is in the form of a particular vector
space called Hilbert space ; the structure of this kind of vector space imposes
a set of constraints on its elements so that divergence problems, such as
the one we just mentioned, no longer bother us. When we embed infinite
sequences into a Hilbert space, these constraints translate to the condition
that the corresponding signals have finite energy - which is a mild and rea-
sonable requirement.
Finally, it is important to remember that the notion of Hilbert space is
applicable to much more general vector spaces than

N ;forinstance,we
can easily consider spaces of functions over an interval or over the real line.
This generality is actually the cornerstone of a branch of mathematics called
functional analysis . While we will not follow in great depth these kinds of
generalizations, we will certainly point out a few of them along the way.
The space of square integrable functions, for instance, will turn out to be
a marvelous tool a few Chapters from now when, finally, the link between
continuous—and discrete—time signals will be explored in detail.

3.2.1 The Recipe for Hilbert Space

A word of caution: we are now starting to operate in a world of complete
abstraction. Here a vector is an entity per se and, while analogies and ex-
amples in terms of Euclidean geometry can be useful visually, they are, by
no means, exhaustive. In other words: vectors are no longer just N -tuples
of numbers; they can be anything. This said, a Hilbert space can be defined
in incremental steps starting from a general notion of vector space and by
supplementing this space with two additional features: the existence of an
inner product and the property of completeness.

Vector Space. Consider a set of vectors V and a set of scalars S (which
can be either

is completely
defined by the existence of a vector addition operation and a scalar multi-
plication operation which satisfy the following properties for any x , y , z ,

or for our purposes). A vector space H ( V , S )

∈ V

and any α , β ∈ S :