This is a 50 minute lecture in a engineering class; one can easily see the mathematical demands put on the students. Many of the seemingly abstract facts from calculus (differentiability, continuity, convergence of a sequence of functions) are heavily used. Of particular interest to me is the remarks from 45 to 50 minutes into the video:
Here is what is going on: if we have a sequence of functions defined on some interval and if is defined on , then we say that “in mean” (or “in the norm”). Basically, as grows, the area between the graphs of and gets arbitrarily small.
However this does NOT mean that converges to point wise!
If that seems strange: remember that the distance between the graphs can say fixed over a set of decreasing measure.
Here is an example that illustrates this: consider the intervals The intervals have length and start by moving left to right on and then moving right to left and so on. They “dance” on [0,1]. Let the the function that is 1 on the interval and 0 off of it. Then clearly as the interval over which we are integrating is shrinking to zero, but this sequence of functions doesn’t converge point wise ANYWHERE on . Of course, a subsequence of functions converges pointwise.