## Scattered comments #1: PMATH 753 pt. 1

As I mentioned last post, my camera is currently out of service, so I gave up and decided I would, for the time being, write a series of posts discussing/clarifying lecture material and assignments from my pure math courses this term (PMATH 442, PMATH 733, and PMATH 753). This post is the first of a trilogy, and discusses PMATH 753. I will generally discuss things in chronological order, although this is not completely possible since assignments usually “look ahead” a bit into the lecture material. That being said, some things may seem out of order. I really worked hard on this post, as I will on the others.

PMATH 753: Functional Analysis [Assignment 1 here]

Functional analysis, all right! In this course we study vector spaces and linear maps. That’s right, this stuff is basically grounded in linear algebra; the only catch is that we care about the ones that are normed (which means they carry nice metric structure on which we can do analysis) and complete, which are called Banach spaces. Functional analysis focuses mostly on infinite-dimensional Banach spaces (and more generally, “nice” families of topological vector spaces) and the continuous linear maps between them. As occult as this may sound, there is rather compelling motivation for its study: the classical sequence and function spaces $\ell_p$ and $L_p$ are Banach spaces! Of course, the familiar $\mathbb{R}^n$ and $\mathbb{C}^n$ are (finite-dimensional) Banach spaces too, and their study is by no means trivial either. From what I’ve read, elements of functional analysis, ranging all the way from basic Hilbert space theory to more advanced topics like C*-algebras, are applied heavily in theoretical/mathematical physics as well (e.g. in the formulation of quantum mechanics), but I don’t know many details on that currently.

In the first week, we reviewed concepts of normed vector spaces, Banach spaces, and topological spaces, familiar from such courses as PMATH 351 (Real Analysis) and PMATH 450 (Lebesgue Integration and Fourier Analysis), and saw some examples (the Sorgenfrey line is cool: an example of a non-metrizable topology on $\mathbb{R}$). We also reviewed the topological definition of continuity, and A1P1 showed it was equivalent to other common formulations.

$\mathbb{F}$ denotes $\mathbb{R}$ or $\mathbb{C}$. We saw, for a topological space $(X,\tau)$, the space $\mathcal{C}_b(X,\tau) = \mathcal{C}_b^{\mathbb{F}}(X,\tau)$ of continuous bounded functions $X \to \mathbb{F}$, and showed it was complete under the supremum norm $\| \cdot \|_\infty$, which really only hinged on the completeness of $\mathbb{F}$, hence we were able to generalize this to the analogous space $\mathcal{C}_b^{\mathcal{Y}}(X,\tau)$ where the codomain is a Banach space $\mathcal{Y}$ rather than $\mathbb{F}$. This space and its completeness becomes (perhaps surprisingly) useful to us, for a few reasons:

• The concepts of “boundedness on the unit disc $\mathsf{D}(X)$” and “continuity” coincide for linear maps (indeed, we even get Lipschitz continuity, which is rather pleasing).
• If you know the value of a linear map on any arbitrarily small disc (any arbitrarily small sphere, actually, since linear maps send 0 to 0), you know it everywhere, just by scaling those values appropriately. This comes as no surprise, because even stronger statements can be made: a linear map is completely determined by its values on a (Hamel) basis, and a continuous linear map is completely determined by its values on a Schauder basis.

This is nicer than you think. For one thing, these two observations allowed us to prove that, for $X$ normed and $\mathcal{Y}$ Banach, the space $\mathcal{B}(X,\mathcal{Y})$ of bounded linear maps $X \to \mathcal{Y}$ is complete. To do this, we used the two observations above to establish an isometric linear isomorphism $\Gamma : \mathcal{B}(X, \mathcal{Y}) \to \mathcal{C}_b^{\mathcal{Y}}(\mathsf{B}(X))$, namely, $\Gamma$ takes a linear map $X \to \mathcal{Y}$ and restricts it to the unit ball $\mathsf{B}(X)$. What do you know, the operator norm on $\mathcal{B}(X, \mathcal{Y})$ coincides with the sup-norm on $\mathcal{C}_b^{\mathcal{Y}}(\mathsf{B}(X))$. Hmm, the definition of “bounded linear map” suddenly seems all the more cunning. As an immediate consequence of this, we see that the continuous dual $X^* = \mathcal{B}(X, \mathbb{F})$ of any normed space is complete. A1P2, in requiring us to establish an isometric linear isomorphism $\ell_p \cong {\ell_q}^*$, lead to a quick and natural proof that $\ell_p$ was complete: it “is” a dual space! The completeness of dual spaces crops up later, too: roughly speaking, because duals are complete, and vector spaces admit a natural isomorphism to their double duals, the completion of a normed space $X$, i.e. the “Banachization” of $X$, lives naturally inside $X^{**}$. This is actually a corollary of the Hahn-Banach theorem, which I will talk about later.

Furthermore, it is quite comforting to know that if two Banach spaces $\mathcal{X}$ and $\mathcal{Y}$ are not isometrically isomorphic, then there is some intrinsic difference in the geometry of their unit spheres. Indeed, this was arguably the crux of the first bonus problem, A1P3d, which asked whether the spaces $(\pmb{c}, \| \cdot \|_\infty)$ and $(\pmb{c}_0, \| \cdot \|_\infty)$ (of convergent sequences, and convergent-to-0 sequences, respectively) were isometrically isomorphic. If you made this observation, you would begin thinking about their unit spheres. Perhaps you might be tempted to first consider compactness, but the unit ball of an infinite-dimensional normed space is never compact (c.f. Riesz’ lemma), and moreover you’ll probably have more luck if you seek a metric difference, not a topological one (troll comment: I’ve since learned that these two spaces are topologically isomorphic LOL). Therefore, you might subsequently consider convexity, which would naturally lead you to thinking about extreme points, which aim at the heart of the problem. It turns out that any sequence in $\pmb{c}$ which consists only of $+1$s and $-1$s is an extreme point of the unit sphere, so $\mathsf{S}(\pmb{c})$ has $2^{\aleph_0}$ (uncountably many) extreme points (update: oh wait, but the sequence has to eventually become constant, so there are only countably many such points). On the other hand, the unit sphere of $\pmb{c}_0$ admits no extreme points, because the convergence-to-0 behaviour guarantees us that for a sequence $x=(x_k) \in \mathsf{S}(\pmb{c}_0)$ the following lovely (but patently obvious) things occur:

• The supremum is actually achieved by some entry, i.e. $|x_\ell| = 1$ for some $\ell$.
• There is some “breathing room”, i.e. $|x_k| < 1$ for some $k$ (clearly $k \neq \ell$).

Using the two observations above, we note that $x=(y+z)/2$ where $y$ and $z$ are obtained merely by adding and subtracting, respectively, some small $\epsilon$ from $x_k$. Clearly, since we have not messed with $x_\ell$, we have $y, z \in \mathsf{S}(\pmb{c}_0)$. So there are no extreme points of that sphere. The problem is then solved by noting that any isometric linear isomorphism would map one unit sphere onto the other while preserving extreme points. By the way, because the spheres of both spaces admit points that are non-extreme, neither $\pmb{c}$ nor $\pmb{c}_0$ is strictly convex. However, any inner product space is.

Speaking of strict convexity, at the end of the first week, we talked about strictly convex maps $\varphi : \mathbb{R} \supseteq I \to \mathbb{R}$, and established an AM-GMish inequality that proved invaluable in the proof of Hölder’s inequality. We then exploited Hölder’s inequality to prove Minkowski’s inequality, which is the $p$-norm triangle inequality. Hence by limiting arguments, we established that $\ell_p$ is a normed space (we talked about its completeness above).

The rest of A1 mostly consisted of slick ways of contriving sequences using the signum function $\mathrm{sgn}(re^{it}) = e^{it}$. Density arguments came in handy quite often, particularly the space $\ell$ (more commonly denoted $c_{00}$) of sequences with cofinitely many entries equal to 0, equipped with a suitable norm. For example, my solution to A1P3b proved that $\| f_y \| \leq \| y \|_1$ for all $y \in \ell_1$, $\| f_y \| \geq \| y \|_1$ for all $y \in c_{00} \subset \ell_1$, and then noted that the first inequality establishes the continuity of the obviously linear map $\ell_1 \to \pmb{c}(\mathbb{Z})^*$ given by $y \mapsto f_y$, allowing us to conclude that indeed $\| f_y \| = \| y \|_1$ for all $y \in \ell_1$ as required. A1P4 showed us that the dual of $\ell_\infty$ “is” the space $\mathcal{FA}(\mathbb{N})$ (more commonly denoted $ba(2^\mathbb{N})$) of finitely additive set functions of bounded variation on $\mathbb{N}$. Sadly, I didn’t have time to solve A1P4e, the second bonus problem, which essentially is concerned with showing that ${\ell_\infty}^*$ is not $\ell_1$. Given the time, I would have attempted to argue this based on the fact that $\ell_\infty$ (and hence its dual) are not separable, but perhaps someone knows of a more clever solution. Also, A1P4a has a rather trivial solution if one notes that $x(\mathbb{N})$ is totally bounded and therefore admits a finite $\epsilon$-net for any $\epsilon$ (this is PMATH 351 material).

The first lecture of the second week (Lecture 4) was filled with hints for A1, most of which I already discussed above. Lecture 5 introduced the Hahn-Banach theorem, debatably the first true hammer of functional analysis

…which I can certainly not do justice to in the tail end of a blog post that is already far too lengthy as it is. This “trilogy” might actually end up being an endless sequence of posts.