Today we talk tensor products. Specifically this post covers the construction of the tensor product between two modules over a ring. But before jumping in, I think now's a good time to ask, "What are tensor products good for?" Here's a simple example where such a question might arise...

This post is the third example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence of continuous functions which converges in the $L^1$ norm (the set of Lebesgue measurable functions), but does not converge uniformly.

This post is the second example in an ongoing list of various sequences of functions which converge to different things in different ways. Here's a sequence which converges uniformly but does not converge in $L^1$ (the set of Lebesgue measurable functions).

Given a sequence of real-valued functions $\{f_n\}$, the phrase, "$f_n$ converges to a function $f$" can mean a few things:

• $f_n$ converges uniformly
• $f_n$ converges pointwise
• $f_n$ converges almost everywhere (a.e.)
• $f_n$ converges in $L^1$ (set of Lebesgue integrable functions)
• and so on...

Other factors come into play if the $f_n$ are required to be continuous, defined on a compact set, integrable, etc.. So since I do not have the memory of an elephant (whatever that phrase means...), I've decided to keep a list of different sequences that converge (or don't converge) to different functions in different ways. With each example I'll also include a little (and hopefully) intuitive explanation for why. Having these sequences close at hand is  especially useful when analyzing the behavior of certain functions or constructing counterexamples.

In any area of math, it's always good idea to keep a few counterexamples in your back pocket. This post continues part 1 with examples/non-examples from some of the different subsets of integral domains.