# August 2015

# Completing a Metric Space, Intuitively

An incomplete metric space is very much like a golf course: it has a lot of missing points!

# What's a Transitive Group Action?

Let a group $G$ act on a set $X$. The action is said to be transitive if for any two $x,y\in X$ there is a $g\in G$ such that $g\cdot x=y$. This is equivalent to saying there is an $x\in X$ such that $\text{orb}(x)=X$, i.e. *there is exactly one orbit*. And all this is just the fancy way of saying that $G$ shuffles all the elements of $X$ *among themselves*. In other words…

# One Unspoken Rule of Measure Theory

Here's a measure theory trick: when asked to prove that a set of points in $\mathbb{R}$ (or some measure space $X$) has a certain property, try to show that the set of points which does **NOT** have that property has measure 0! This technique is used quite often.

# Two Ways to be Small

In real analysis, there are two ways a *measurable set* $E$ can be small. Either

- the measure of $E$ is 0, OR
- $E$ is
*nowhere dense.*

Intuitively, to say the measure of $E$ is $0$ means that...

# Need to Prove Your Ring is NOT a UFD?

You're given a ring $R$ and are asked to show it's *not* a UFD. Where do you begin? One standard trick is to apply the Rational Roots Theorem….

# Operator Norm, Intuitively

If $X$ and $Y$ are normed vector spaces, a linear map $T:X\to Y$ is said to be <b>bounded</b> if $\|T\|< \infty$ where

$$\|T\|=\sup_{\underset{x\neq 0}{x\in X}}\left\{\frac{|T(x)|}{|x|}\right\}.$$

(Note that $|T(x)|$ is the norm *in* $Y$ whereas $|x|$ is the norm *in* $X$.) One can show that this is equivalent to

$$\|T\|=\sup_{x\in X}\{|T(x)|:|x|=1\}.$$ So intuitively (at least in two dimensions), we can think of $\|T\|$ this way…

# Borel-Cantelli Lemma (Pictorially)

The Borel-Cantelli Lemma says that if $(X,\Sigma,\mu)$ is a measure space with $\mu(X)<\infty$ and if $\{E_n\}_{n=1}^\infty$ is a sequence of measurable sets such that $\sum_n\mu(E_n)<\infty$, then $$\mu\left(\bigcap_{n=1}^\infty \bigcup_{k=n}^\infty E_k\right)=\mu\left(\limsup_{n\to\infty} En \right)=0.$$

(For the record, I didn't understand this when I first saw it (or for a long time afterwards). My only thought was, "But what does that *mean? In English??*") To help our intuition…

# What do Polygons and Galois Theory Have in Common?

*Galois Theory is all about symmetry.* So, perhaps not surprisingly, symmetries found among the **roots of polynomials** (via Galois theory) are closely related to symmetries of **polygons in the plane** (via geometry). In fact, the two are highly analogous!

# One Unspoken Rule of Algebra

Here's an algebra tip! Whenever you're asked to prove $$A/B\cong C$$ where $A,B,C$ are groups, rings, fields, modules, etc., *mostly likely* the The First Isomorphism Theorem involved!

# Motivation for the Tensor Product

In general, if $F$ is a field and $V$ is a vector space over $F$, the tensor product answers the question "How can I define scalar multiplication on $V$ by some *larger* field which contains $F$?" (Of course this holds if we replace the word "field" by "ring" and consider the same scenario with modules.)

# Why are Noetherian Rings Special?

In short, "Noetherian-ness" is a property which generalizes "PID-ness." As Keith Conrad so nicely puts it, "The property of all ideals being singly generated is often not preserved under common ring-theoretic constructions (e.g. $\mathbb{Z}$ is a PID but $\mathbb{Z}[x]$ is not), but the property of all ideals being finitely generated *does* remain valid under many constructions of new rings from old rings. For example... every quadratic ring $\mathbb{Z}[\sqrt{d}]$ is Noetherian, even though many of these rings are not PIDs." (italics added)

# What is Galois Theory Anyway?

Perhaps you've heard of Évariste Galois? (Pronounced "GAL-wah.") You know, the French mathematician who died tragically in 1832 in a duel at the tender age of 20? (Supposedly over a girl! C*'est romantique, n'est-ce pas?*) Well, today we're taking a bird's-eye view of his most well-known contribution to mathematics: the appropriately named Galois theory. The goal of this post is twofold...

# Baire Category & Nowhere Differentiable Functions (Part Two)

Welcome to part two of our discussion on Baire's Category Theorem. Today we'll sketch the proof that we can find a continuous function on $[0,1]$ which is *nowhere* differentiable.

# A Non-Measurable Set

Today we're looking at a fairly simple proof of a standard result in measure theory:

**Theorem**: Any measurable subset $A$ of the real line with positive measure contains a non-measurable subset.

(Remark: we used this theorem last week to prove the existence of a Lebesgue measurable set which is not a Borel set.)

# Lebesgue Measurable But Not Borel

Our goal for today is to construct a Lebesgue measurable set which is *not *a Borel set. In summary, we will define a homeomorphism from $[0,1]$ to $[0,2]$ which will map a (sub)set (of the Cantor set) of measure 0 to a set of measure 1. This set of measure 1 contains a non-measurable subset, say $N$. And the preimage of $N$ will be Lebesgue measurable but will not be a Borel set.

# Baire Category & Nowhere Differentiable Functions (Part One)

The Baire Category Theorem is a powerful result that relates a metric space to its underlying topology. (And sadly no, nothing to do with category theory!) Informally, the theorem says that if you can find a metric with respect to which your topological space is complete, then that space cannot be written as a countable union of nowhere dense sets. In other words, a metric can put a restriction on the topology.