In sum, the sufficient condition (a.k.a. the "if" direction) allows you to get what you want. That is, if you assume the sufficient condition, you'll obtain your desired conclusion. It's enough. It's sufficient. 

On the other hand, the necessary condition (a.k.a. the "only if" direction) is the one you must assume in order to get what you want. In other words, if you don't have the necessary condition then you can't reach your desired conclusion. It is necessary. 

Welcome to the final post in a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. Today we prove two lemmas (the path- and homotopy-lifting properties) that were used in parts four and five. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Below is a general method —a recipe, if you will —for computing the universal cover of the wedge sum $X\vee Y$ of arbitrary topological spaces $X$ and $Y$. This is simply a short-and-quick guideline that my prof mentioned in class, and I thought it'd be helpful to share on the blog. To help illustrate each step, we'll consider the case when $X=T^2$ is the torus and $Y=S^1$ is the circle.       

Welcome to part five of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is injective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Have you ever read Tristan Needham’s Visual Complex Analysis? I highly recommend this book as a supplement to a standard undergrad/grad course in complex analysis. It's nothing (nothing!) like your usual textbook. The author writes to build your intuition and insight, so it's warm like a conversation and not cold like some math texts. It’s also loaded with illustrations (hence the title), historical background, and context. For example, did you know

Welcome to part four of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our homomorphism from $\mathbb{Z}$ to $\pi_1(S^1)$ is surjective. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Welcome to part three of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we prove that our map from $\mathbb{Z}$ to $\pi_1(S^1)$ is a group homomorphism. The proof follows that found in Hatcher's Algebraic Topology section 1.1.

Welcome to part two of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we justify a shortcut that we never actually use in the remainder of this series, so the reader is welcome to skip this post. But I've included it since, in this series, we're closely following section 1.1 of Hatcher's Algebraic Topology.

Welcome to part one of a six-part series where we prove that the fundamental group of the circle $\pi_1(S^1)$ is isomorphic to $\mathbb{Z}$. In this post we define a map from $\mathbb{Z}$ to $\pi_1(S^1)$ and make some simple observations via pictures and an animation! The proof follows that found in Hatcher's Algebraic Topology</a>, section 1.1.

Up to isomorphism” is a phrase that seems to get thrown around a lot without ever being explained. Simply put, we say two groups (or any other algebraic structures) are the same “up to isomorphism” if they’re isomorphic! In other words, they share the exact same structure and therefore they are essentially indistinguishable. Hence we consider them to be one and the same! But, you see, we mathematicians are very precise, and so we really don't like to use the word “same." Instead we prefer to say “same up to isomorphism.” Voila!

The quote above comes from an excellent Slate article by Chase Felker on why students shouldn't be afraid of or intimidated by mathematics. I posted the quote on Instagram not too long ago, and since it addresses a topic that is near-and-dear to my own heart, I decided to include it on the blog as well. Felker prefaces the quote by saying, "Giving up on math means you don't believe that careful study can change the way you think."

Here's a chart to help keep track of some of the different "flavors" of continuity in real analysis.

In this post, we visually explore the definition of a transitive group action and see how it relates to the phrase, "Where there's a will, there's a way!"

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$, answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Dominated Convergence Theorem and see why "domination" is necessary.

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss the Monotone Convergence Theorem and solve a nasty-looking problem which, thanks to the theorem, is actually quite trivial!

Have you read Jeffrey Weeks' The Shape of Space before? What a great book! It explores the geometry of spheres, tori, Möbius strips, Klein bottles, projective planes and other spaces in an engaging, this-is-definitely-not-a-textbook kind of way. Other topics include: gluing, orientability, connected sums, Euler number, hyperspace, bundles, and more! (Have I whet your appetite yet?!)

Fatou's Lemma, the Monotone Convergence Theorem, and the Dominated Convergence Theorem are three major results in the theory of Lebesgue integration which, when given a sequence of functions $\{f_n\}$ answer the question, "When can I switch the limit symbol and the integral symbol?" In this post, we discuss Fatou's Lemma and solve a problem from Rudin's Real and Complex Analysis (a.k.a. "Big Rudin").

Today we're chatting about the Borel-Cantelli Lemma. When I first came across this lemma, I struggled to understand what it meant "in English." What does $\mu(\cup\cap E_k)=0$ really signify?? There's a pretty simple explanation if $(X,\Sigma,\mu)$ is a probability space, but how are we to understand the result in the context of general measure spaces?

In today's post I want to emphasize a simple - but important - idea in topology which I think is helpful for anyone new to the subject, and that is: Open sets are everything! What do I mean by that? Well, for a given set $X$, all the properties of $X$ are HIGHLY dependent on how you define an "open set."

Today's post is a bit of a ramble, but my goal is to uncover the intuition behind one of the definitions of a connected topological space. Ideally, this is just a little tidbit I'd like to stash in The Back Pocket. But as you can tell already, the length of this post isn't so "little"! Oh well, here we go!

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...

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.

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)

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.)

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.)

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!

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.

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…

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! 

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…

  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...

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…

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….

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

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.

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.

A function $f$ is said to be continuous if the preimage of any open set is open. Analogously, we might say that a function is measurable if the preimage of a measurable set is measurable. It's not hard to show that if $\{f_n\}$ is a sequence of measurable functions, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are also measurable functions. But here the analogy between continuity and measurability breaks down. It is not true that if each $f_n$ is a continuous function, then sup$\{f_n\}$, inf$\{f_n\}$, limsup $\{f_n\}$ and liminf $\{f_n\}$ are continuous as well. Below is a counterexample - a sequence of continuous functions with a discontinuous supremum!

Given a countable collection of measurable sets, is it possible to construct a new collection of sets which are pairwise disjoint and have the same union as the original? Yes! Here's the trick....

This post is the sixth 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 on $[0,1]$ which converges to 0 in $L^1$, but does not converge anywhere on $[0,1]$.

This week we continue our discussion on the Stone Weierstrass Theorem with an example. This exercise is taken from Rudin's Principles of Mathematical Analysis (affectionately known as "Baby Rudin"), chapter 7 #20.

The Stone Weierstrass Theorem is a generalization of the familiar Weierstrass Approximation Theorem. In this post, we introduce the Stone Weierstrass Theorem and, by looking at counterexamples, discover why each of the hypotheses of the theorem are necessary.

Last week we saw an example of how to use the rational canonical form (RCF) to classify matrices of a given order in $GL_2(\mathbb{Q})$. Today we have a similar example (taken from CUNY's spring 2015 qualifying exam) where now our matrices have entires in the finite field $F_13$. The fact that our field is $F_13$ instead of $\mathbb{Q}$ actually makes little difference in how to approach the solution, but I think this problem is particularly nice because part of it calls on some Galois Theory.

Last time we discussed the rational canonical form (RCF) of a linear transformation, and we mentioned that any two similar linear transformations have the same RCF. It's this fact which allows us to classify distinct linear transformations on a given $F$-vector space $V$ for some field $F$. Today, to illustrate this, we'll work through a concrete example:

Find representatives for the distinct conjugacy classes of matrices of finite order in the multiplicative group of 2x2 matrices with rational entries.

This post is intended to be a hopefully-not-too-intimidating summary of the rational canonical form (RCF) of a linear transformation. Of course, anything which involves the word "canonical" is probably intimidating no matter what. But even so, I've attempted to write a distilled version of the material found in (the first half of) section 12.2 from Dummit and Foote's Abstract Algebra.

In sum, the RCF is important because it allows us to classify linear transformations on a vector space up to conjugation. Below we'll set up some background, then define the rational canonical form, and close by discussing why the RCF looks the way it does. Next week we'll go through an explicit example to see exactly how the RCF can be used to classify linear transformations.

We know what it means to have a module $M$ over a (commutative, say) ring $R$. We also know that if our ring $R$ is actually a field, our module becomes a vector space. But what happens if $R$ is "merely" a PID? Answer: A lot. Today we'll look at a proposition, which, thanks to the language of exact sequences, is quite simple and from which the Fundamental Theorem of Finitely Generated Modules over a PID follows almost immediately. The information below is loosely based on section 12.1 of Dummit and Foote' Abstract Algebra.

Today we've got a little post on a little fact relating to group actions. I wanted to write about this not so much to emphasize its importance (it's certainly not a major result) but simply to uncover the intuition behind it.

Today we're talking about Eisenstein's (not Einstein's!) Criterion - a well-known test to determine whether or not a polynomial is irreducible. In particular, we'll consider two examples where a not-so-obvious trick is needed in order to apply the criterion.

When I was first introduced to Noetherian rings, I didn't understand why my professor made such a big hoopla over these things. What makes Noetherian rings so special? Today's post is just a little intuition to stash in The Back Pocket, for anyone hearing the word "Noetherian" for the first time. 

You know the Sylow game. You're given a group of a certain order and are asked to show it's not simple. But where do you start? Here are four options that may be helpful when trying to produce a nontrivial normal subgroup.

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.

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. 

Here is a list of some of the subsets of integral domains, along with the reasoning (a.k.a proofs) of why the bullseye below looks the way it does. Part 2 of this post will include back-pocket examples/non-examples of each.   

The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated." In a Hausdorff space, it's guaranteed that if you pick any two distinct points in the space -- say $x$ and $y$ -- then you can always find an open set containing $x$ and an open set containing $y$ such that those two sets don't overlap.

After some exposure to group theory, you quickly learn that when trying to prove a group $G$ is abelian, checking if $xy=yx$ for arbitrary $x,y$ in $G$ is not always the most efficient - or helpful! - tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:

Yes! I'm writing about math. No! Don't close your browser window. Hear me out first...

I know very well that math has a bad rap. It's often taught or thought of as a dry, intimidating, unapproachable, completely boring, who-in-their-right-mind-would-want-to-think-about-this-on-purpose kind of subject. I get it. Math was the last thing on earth I thought I'd study. Seriously.