Function spaces
During the week of April 2-4, we look at function spaces. Here is a list of
function spaces:
| C∞ | The space of smooth functions |
| C | The space of continuous functions |
| Pn | The space of polynomials of degree less or equal than n. |
| PC∞ | The space of piecewise smooth function |
A blackboard from a class of Thursday, April 4, 2019: Click for larger picture.
It shows also examples of function spaces which are not linear. There is no official name
for the space of piecewise smooth functions. It is denoted F in the lecture.
One could also call it PC
∞. This space is often used when
discussing Fourier series. To make the definition precise, one can define the
space as the set of functions which are smooth everywhere except for finitely
many points, where the function can have a jump discontinuity. We don't allow
poles like 1/x, nor oscillatory discontinuities like sin(1/x).
The next blackboard features also an integration by parts problem
using the Tic-Tac-Toe method. See
this handout (PDF) from 2014.
Or
the tic-tac-toe method seen in the movie Stand and Deliver.
