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## Banach spaces (and Lawvere metrics, and closed categories)

One annoying feature of the abstract theory of vector spaces, and one that often trips up beginners, is that it is not possible to make sense of an infinite sum of vectors in general. If we want to make sense of infinite sums, we should probably define them as limits of finite sums, so rather than work with bare vector spaces we need to work with topological vector spaces over a topological field, usually $\mathbb{R}$ or $\mathbb{C}$ (but sometimes fields like $\mathbb{Q}_p$ are also considered, e.g. in number theory). Common and important examples include spaces of continuous or differentiable functions.

Today we’ll discuss a class of topological vector spaces which is convenient to work with but which still covers many examples of interest, namely Banach spaces. The material in the first half of this post is completely standard and can be found in any text on functional analysis.

In the second half of the post we discuss a category of Banach spaces such that two Banach spaces are isomorphic in this category if and only if they are isometrically isomorphic but which still allows us to talk about bounded linear operators between Banach spaces, and to do this we briefly discuss Lawvere metrics; this material can be found on the nLab.

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## ab, ba, and the spectrum

Let $a, b$ be two $n \times n$ matrices. If $a, b$ don’t commute, then $ab \neq ba$; however, the two share several properties. If either $a$ or $b$ is invertible, then $ab$ is conjugate to $ba$, so in particular they have the same characteristic polynomial.

What if neither $a$ nor $b$ are invertible? As it turns out, $ab$ and $ba$ still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have $ab = 0$ but $ba$ nonzero). There are several ways of proving this result, which implies in particular that $ab$ and $ba$ have the same eigenvalues.

What if $a, b$ are linear transformations on an infinite-dimensional vector space? Do $ab$ and $ba$ still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”

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## Structures on hom-sets

Suppose I hand you a commutative ring $R$. I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of $R$, and you also can’t refer directly to the multiplication or addition maps $R \times R \to R$, since these aren’t morphisms.) Geometrically, I might equivalently say that you are only allowed to work in the language of the category of affine schemes, since the two are dual. Can you recover $R$ as a set, and can you recover the ring operations on $R$?

The answer turns out to be yes. Today we’ll discuss how this works, and along the way we’ll run into some interesting ideas.

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## Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter.

Terence Tao has already written a great introduction to ultrafilters with an eye towards nonstandard analysis. I’d like to introduce them from a different perspective. Some of the topics below are also covered in these posts by Todd Trimble.

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## Zeta functions, statistical mechanics and Haar measure

An interesting result that demonstrates, among other things, the ubiquity of $\pi$ in mathematics is that the probability that two random positive integers are relatively prime is $\frac{6}{\pi^2}$. A more revealing way to write this number is $\frac{1}{\zeta(2)}$, where $\displaystyle \zeta(s) = \sum_{n \ge 1} \frac{1}{n^s}$

is the Riemann zeta function. A few weeks ago this result came up on math.SE in the following form: if you are standing at the origin in $\mathbb{R}^2$ and there is an infinitely thin tree placed at every integer lattice point, then $\frac{6}{\pi^2}$ is the proportion of the lattice points that you can see. In this post I’d like to explain why this “should” be true. This will give me a chance to blog about some material from another math.SE answer of mine which I’ve been meaning to get to, and along the way we’ll reach several other interesting destinations.

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## The adjoint functor theorem for posets

Recently in Measure Theory we needed the following lemma.

Lemma: Let $g : \mathbb{R} \to \mathbb{R}$ be non-constant, right-continuous and non-decreasing, and let $I = (g(-\infty), g(\infty))$. Define $f : I \to \mathbb{R}$ by $f(x) = \text{inf} \{ y \in \mathbb{R} : x \le g(y) \}$. Then $f$ is left-continuous and non-decreasing. Moreover, for $x \in I$ and $y \in \mathbb{R}$, $f(x) \le y \Leftrightarrow x \le g(y)$.

If you’re categorically minded, this last condition should remind you of the definition of a pair of adjoint functors. In fact it is possible to interpret the above lemma this way; it is a special case of the adjoint functor theorem for posets. Today I’d like to briefly explain this. (And who said category theory isn’t useful in analysis?)

The usual caveats regarding someone who’s never studied category talking about it apply. I welcome any corrections.

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## The induced representation

Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up.

Continuing the previous post, what we want to do now is to think of restriction $\text{Res}_H^G : \text{Rep}(G) \to \text{Rep}(H)$ as a forgetful functor, since restricting a representation just corresponds to forgetting some of the data that defines it. Its left adjoint, if it exists, should be a construction of the “free $G$-representation” associated to an $H$-representation. Given a representation $\rho : H \to \text{Aut}(V)$ we therefore want to find a representation $\rho' : G \to \text{Aut}(V')$ with the following universal property: any $H$-intertwining operator $\phi : V \to W$ for $\tau$ a $G$-representation on $W$ naturally determines a unique $G$-intertwining operator $\phi' : V' \to W$. In other words, we want to construct a functor $\text{Ind}_G^H : \text{Rep}(H) \to \text{Rep}(G)$ such that $\text{Hom}_{\text{Rep}(G)}(\text{Ind}_H^G \rho, \tau) \simeq \text{Hom}_{\text{Rep}(H)}(\rho, \text{Res}_H^G \tau)$.

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## Some adjoint functors

Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better.

One way to define a subgroup $H$ of a group $G$ is as the image of a homomorphism into $G$. Given the inclusion map $H \to G$, the functor $\text{Hom}(G, \text{End}(V))$ in the category of groups acts contravariantly to give a map $\text{Res}_H^G : \text{Rep}(G) \to \text{Rep}(H)$ called restriction. More concretely, the restricted representation $\rho|_H$ of a representation $\rho$ is defined simply by $\rho|_H(h) = \rho(h)$. Hence there is a functorial way to pass from a representation of a group $G$ to one of a subgroup $H$.

It is not obvious, however, whether there is a functorial way to pass from a representation of $H$ back to one of $G$. There is such a construction, which goes by the name of induction, and we will need it later. Today we’ll discuss the general category-theoretic context in which induction is understood, where it is called an adjoint functor. For more about adjoints, see (in no particular order) posts at Concrete Nonsense, the Unapologetic Mathematician, and Topological Musings.

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## The orthogonality relations for representations of finite groups

In order to continue our discussion of symmetric functions it will be useful to have some group representation theory prerequisites, although I will use many of the results in the representation theory of the symmetric groups as black boxes. I had planned on using this post to discuss Frobenius reciprocity, but got so carried away with motivating it that this post now stands alone.

Today I’d like to discuss the representation theory of finite groups over $\mathbb{C}$. As these are strong assumptions, the resulting theory is quite elegant, but I always found the proofs a little unmotivated, so I’m going to try to use the categorical perspective to fix that. Admittedly, I don’t have much experience with this kind of thing, so this post is for my own benefit as much as anyone else’s. The main focus of this post is motivating the orthogonality relations.

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