|C[a,b]||Continuous functions on [a,b]|
|L2(a,b)||Square integrable functions:|
|Lp(a,b)||p-integrable functions ():|
|H1(a,b)||Sobolev space: and||||f||H12=||f(x)||L22+||f'(x)||L22|
In the abstract, we will be dealing with a set of functions X with a norm .So far, we don't know what to do with limits of objects xk in X.
If we were talking about real numbers, we would expect that a sequence like 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...would converge. If our universe only consisted of rational numbers, then we would be in trouble because they could be converging to , which is not rational. But with real numbers, any sequence like 1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ..., which adds one new digit with each element in the sequence, converges.
What sequences xk in a set X with norm ``ought'' to converge? The answer that we use is the idea of a Cauchy sequence:
A sequence xk is a Cauchy sequence if for every there is an N such that implies .
A space (consisting of X with norm ) is complete if every Cauchy sequence has a limit.
The first example of a complete space is the real line.
The first example of a complete function space that most people meet is the space of continuous functions on [a,b], denoted C[a,b], with norm .
All of the examples from §2 are complete function spaces.
Complete normed spaces are called Banach spaces after the Polish mathematician, Stefan Banach.
One of the most important function spaces is L2(a,b) - the space of square integrable functions on the interval (a,b). The L2 norm is .
The precise definition of L2(a,b) is based on Lebesgue integration theory, which is beyond the scope of these notes. The trickiest aspect of L2(a,b) is that we cannot distinguish between functions in L2(a,b) that differ only at a single point, or at a countable number of points.
[Technically, we regard f=g if the set is a null set; that is, if changing the values of this set has no effect on any integrals.]
One of the more important aspects of L2(a,b) is that the norm comes from an inner product:
for real-valued functions. This inner product has the properties that we need for an inner product:
Now we will see the Cauchy inequality holds for these inner-product spaces:
Proof. For any a, , so the quadratic in a,
This means that the quadratic has at most one (repeated) real root. This means that either (g,g)=0, or the discriminant (``b2-4ac'') cannot be positive. In the first case, ||g||=0, so g=0, and there is nothing to show. In the second case, .That is, as required. QED.
This makes L2(a,b) an example of a complete inner product space, or as it is better known, L2(a,b) is a Hilbert space.
Since functions in L2(a,b) are integrable, they represent distributions; L2 is a subset of the space of distributions.
Sobolev spaces are Banach spaces where the norm involves derivatives, or at least, something other than just function values. The simplest of these is H1(a,b).
H1(a,b) is the Banach space of functions
with the norm
||f||H12 = ||f||L22 + ||f'||L22.
Actually, we need to be a little careful about ``'', since we need to keep a tight connection between f and f'. For example, if we think of the Heaviside function H(x) of §1.1, H'(x)=0 for all x, except for x=0, where it is undefined. Thus as far as integrals are concerned, we cannot distinguish between H'(x) and the zero function. On the other hand, H(x) is definitely not constant. So we would like to keep this function out of the Sobolev space H1(a,b).
There are two ways of avoiding functions like H(x):
The Sobolev space H1(a,b) has an inner product that defines the norm:
Higher order Sobolev spaces can be defined:
with the norm
The corresponding inner product is