This post is the fifth example in an ongoing list of various sequences of functions which converge to different things in different ways. Today we have a sequence of functions which converges to 0 pointwise but does not converge to 0 in $L^1$.

Suffixes are important! 

Did you know that the words

"maximal" and "maximum" generally do NOT mean the same thing

in mathematics? It wasn't until I had to think about Zorn's Lemma in the context of maximal ideals that I actually thought about this, but more on that in a moment. Let's start by comparing the definitions:

This post is the fourth example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence of Lebesgue integrable functions which converges uniformly to a function which is not Lebesgue integrable.

When you hear the word closure, what do you think of? I think of wholeness - you know, tying loose ends, wrapping things up, filling in the missing parts. This same idea is behind the mathematician's notion of closure, as in the phrase "taking the closure" of a set. Intuitively this just means adding in any missing pieces so that the result is complete, whole.