To understand the wavelet transform, it helps to have an appreciation of the basic programme of Hilbert space analysis, which is as follows:
Identify the inner product space of interest.
An inner product space (IPS) consists of a (closed) vector space and an
inner product defined on that space.
For example, let V be the space of real-valued functions
with domain the real line (
), and let the inner product
< f, g > be defined for all f, g in V as
Also, f and g are said to be orthogonal when <f,g> = 0.
The norm of a function f is given in terms of the inner
product as 
.
Given the inner product from above, the norm is directly analogous to
a length in Euclidean space.
Using the norm, a Hilbert space 
is defined
to be
It is very natural to let p=2, in which case the space of interest
is said to contain all f that are square-integrable on V.
In the ensuing discussion, we concentrate on the Hilbert space 
.
A linear operator L in a Hilbert space H is one for which
and
for all constants c and all f, g in H.
Usually this operator is specified in two parts. The first is a linear
differential expression 
. The second part is a separately
specified domain of the operator Dom(
), arising from
closure under the operation 
. This closure defines the
maximal operator, which may be further restricted by the
imposition of certain boundary conditions.
It does no harm to repeat that the inner product as used above must be conjugate bilinear:
Hermitian symmetric:
and positive definite:
These properties of the general inner product can be used to prove Schwarz's inequality:
and the triangle inequality:
for any inner product space.
Finally, a mean square metric can be defined in analogy to the
geometric distance between two points in space. The distance between
two functions f and g in an IPS is simply 
.
This metric is used in demonstrating convergence of the expansion of
an arbitrary function in terms of an orthogonal basis as the number of
terms increases.
The eigenvalues of a linear operator L are generally found by solving the equation
to determine the allowed eigenvalues
, and then finding the
associated eigenfunctions
which satisfy the above
equation.
The Hilbert space 
has more than one orthonormal
basis. For example, the Fourier transform can be used to generate
four separate, orthonormal bases of that space. One can also
discover orthogonal bases by determining the eigenspaces of particular
operators. The direct sum of these eigenspaces will be the domain of the
operator, and if the domain of the operator is the whole Hilbert
space, then one has found a basis for the space.
Such arguments are not proven trivially.