Constructing the Tensor Product of Modules

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...

Read More →

On Constructing Functions, Part 3

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.

Read More →

On Constructing Functions, Part 2

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).

Read More →

On Constructing Functions, Part 1

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.

Read More →

The Integral Domain Hierarchy, Part 2

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.

Read More →