Continuous Functions and Hilbert Spaces - Vision with Direction - page 57

4.3 A Scalar Product for Vector Spaces of Functions

Finally, we need to introduce a scalar product of functions :

= f ∗ g f, g (4.5)

· ∗ on the top of g represents the complex conjugate operation. This defi-
nition resembles the scalar product for C N , see Eq. (3.27). Instead of a summation
over a discrete set, we integrate now over a continuum. The integration is taken over
a suitable definition domain of the functions f and g , that is in practice a suitable
range of the arguments of f and g . Integral operation can be seen as a “degenerate”
form of summation when we have so many terms to sum that they are uncountably
infinite. The idea of uncountably infinite needs some more explanation. Integers that
run from 0 to

where the

are, as an example, infinite in number, yet they are countable (nam-
able). That is, there is no integer between two successive integers since all integers
can be named one after the other. By contrast, the real numbers are infinite and un-
countable, because there is always another real number between two real numbers,
no matter how close they are chosen in any imaginable process that will attempt to
name them one after the other. So if we want to make summations over terms that
are generated by real number “indices” (instead of integer indices), we need integra-
tion to sum them up because these “indices” are uncountably infinite. It is possible
to define other scalar products as well, but for most signal analysis applications Eq.
(4.5) will be a useful scalar product.
We can see that if an auto-scalar product of a vector in the Hilbert space is taken
then the result is real and nonnegative. To be more precise,

∞

f, f ≥ 0, and equality

occurs if and only if f =0.

4.4 Orthogonality

With a scalar product we have the tool to test if functions are “orthogonal” in the
same way as before. Two functions are said to be orthogonal if

f, g =0 (4.6)

We can reconstruct and analyze a signal by means of a set of orthogonal basis
signals. Just as a vector in E 3 can be expressed as a weighted sum of three orthog-
onal vectors, so can a function be expressed as a weighted sum of orthogonal basis
functions with some “correctly” chosen coefficients. But in order to do that, we will
need to be more precise about the Hilbert space, i.e., about our scalar product and
functions that are allowed to be in our function space. In Chap. 5 we will give further
details on these matters.