Oct 18 
F 
We looked at sufficient conditions for optimization (Weierstrass theorem).
We realized that two conditions needed in that theorem were both topological issues.
We looked at how those topological concepts were related to the example on the "real" domain.
We looked at open sets and continuity on R, and then generalized to see what general topology is.
This led to two out of the three: closedness, compactness and completeness.
The first two concepts being topological invariants.
We looked at completeness in metric spaces, and then we introduced vector spaces and
we defined Banach spaces as a complete normed vector space.


Nov 1 
F 
This lecture, we will now start looking at function spaces, and study completeness in those spaces.
We will do this in two different directions.
We will look at convergence of functions and see why Riemann integration has some serious flaws
in that context that can be fixed by developing "measure theory".
We will then look at completing function spaces in the same way as rationals are completed into reals.
We will also see how functions can be expanded to accommodate extremely important distributions
that are not functions, such as the Dirac delta.

