Why Functional Analysis


	Many of results from linear algebra of finite dimensional spaces can
be applied to infinite dimension spaces as well. Why infinite dimensions?
Consider the differentiation operator D from the C^oo, the space of
infinitely differentiable functions, to itself given by Df is the function
which takes t |-> f'(t). Big D is a linear operator between vector spaces.

	Linear algebra says Lx = b has a solution if the linear map is onto,
it has a unique solution if the kernel of L is the trivial {0}. And if the
solution is not unique, it decribes the general solution. Furthermore
the inverse map, when it exists, is continuous if you move b a bit, it
moves the solution x a bit. Each of these translate to the oo-dimensional
setting, accept the continuity of the inverse map is no longer automatic.
It also might be more difficult to establish the operator is onto.

	Other features of linear algebra which are worth exploring in oo-dimensions
include duality theory. For example, L is onto <=> L* is 1-1 and L is 1-1
<=> L* is onto. In finite dimensions, both conditions are equivalent. In
oo-dimensions none of these statements is true in general, but there are
interesting special cases. For example, L has dense range <=> L* is 1-1
and L is 1-1 <=> L* has dense range. That is the topology on space manners
in oo-dimensions in way it didn't in finite dimensions. (Perhaps a better
way of saying this is the topology comes for free in finite dimensions.)

	Finally, what would linear algebra be without eigenvalues and eigenvectors.
The big D operator has infinitely many eigenvalues lambda. Since D exp(lambda t)
= lambda exp(lambda t), so eigenvector for lambda is exp(lambda t). This
makes big D an unbounded operator. 



	Why the special cases: Norm spaces, Compact operators, and simple examples


	One, these are a useful subset of the Functional Analysis toolbox. Two,
there is already plenty of variety, both of good examples and pathology. Three,
the most general cases require more background than we need to assume. The
tricky parts we can avoid by noting things like

1. L_1 the space of Lebseque integrable functions, we regard as just the
completion of the Riemann integrable functions.
2. The weak and weak-star topologies are described in terms that don't
require the full definition of topological vector spaces, and instead
look like point-wise convergence.
3. Distributions are done from the applied viewpoint, namely how to use
them rather than from the theoretical viewpoint which needs the topological
vector space theory.
4. Norms are useful even in finite dimensions for numerical work.
5. Function spaces, Differential operators, Metric spaces are all part of
a typical undergraduate mathematical eduation.