## Scattered comments #1: PMATH 753 pt. 2

Time for another contribution to my Scattered Comments series (this is still part of SC1 because the deadline for PMATH 753 A2 has not passed yet). Last time we left off right before Lecture 5 which involved the Hahn-Banach theorem, an innocent-looking theorem for plain old vector spaces that has earth-shattering consequences in the world of normed spaces. As we will see, this theorem is the quintessential wizardry that guarantees us we can separate relatively nice subsets by a hyperplane. This “separation theorem”, the geometric incarnation of Hahn-Banach, tells us some nice things, for example that the convex hull of a set is actually the intersection of all closed half-spaces containing it — anything outside of the convex hull can be “separated” from the rest of the convex hull, essentially by means of a hyperplane, thereby allowing us to “kick it out” of the intersection.

Hahn-Banach Theorem (“HBT”). Let $\mathcal{X}$ be a vector space and $p : \mathcal{X} \to \mathbb{R}$ a sublinear functional (i.e. $p$ is positive homogeneous, and subadditive). If $f \in \mathcal{Y}'$ satisfies $f \leq p$ on $\mathcal{Y}$ then there exists an extension of $f$ to all of $\mathcal{X}$ which is also dominated by $p$ on all of $\mathcal{X}$.

The idea behind the proof is as follows. Without loss of generality (as we will see), we deal only with the case when $\mathcal{Y}$ is of codimension 1 in $\mathcal{X}$, say $\mathcal{X} = \mathrm{span}\{ x, \mathcal{Y} \}$. We show that there’s some number $c$ that squeezes between everything of the form $f(y_-) - p(y_- - x)$ and $p(y_- + x) - f(y_+)$ where $y_-, y_+ \in \mathcal{Y}$. We use this to define $F(\alpha x + y) = \alpha c + f(y)$, then proceed to check it’s a legitimate extension and does what we want it to. The general result follows by “ordering” these extensions $(\varphi, \mathcal{M})$ in the natural way and invoking Zorn’s lemma.

Observe that if $\mathcal{X}$ is normed, then $\| \cdot \|$ is a pretty nice example of a sublinear functional. As a corollary, any continuous functional on a subspace can be extended to a continuous functional (of the same norm!) on the whole space. Another consequence is that (for normed spaces) we get an isometric embedding $\mathcal{X} \hookrightarrow \mathcal{X}^{**}$ into the double-dual. As I mentioned before, upon closing the image of this inclusion, we obtain a special Banach space known as the completion of $\mathcal{X}$. It contains (a copy of) $\mathcal{X}$ as a dense subset.

All this is rather nice and makes us appreciate our norm all the more — indeed, certain topological vector spaces have trivial continuous duals, but HBT guarantees us that our assertive $\| \cdot \|$ will sternly forbid such shenanigans, altruistically reassuring us that our dual space is a fertile garden full of beautiful continuous functionals. On the other hand, if we’re in infinite dimensions, there are some rotten weeds as well: assuming one accepts the existence of Hamel bases, one obtains unbounded (i.e. discontinuous, hence everywhere-discontinuous) linear functionals.

We then moved on, pushing forward towards our goal: separation by hyperplanes. A hyperplane is defined to be “the kernel of a nontrivial linear functional $f$“. We saw that one of two things happen: if the functional $f$ is bounded then its kernel is closed and nowhere dense (as expected) but if it is unbounded, then in fact its kernel is dense (?!). It follows immediately that, in fact, the pullback $f^{-1}(\alpha)$ of any $\alpha \in \mathbb{F}$ is dense. This is pretty hard to picture — kind of reminds me of Picard’s Great Theorem from complex analysis — but it really drives home the fact that the behaviour of such a functional is pathologically disgusting and awful.

The next post will cover convex/absorbing sets and thereafter culminate in a discussion of separation by hyperplanes. After this we will discuss the Baire Category theorem, which is traditionally used to prove such functional-analytic hammers as the open mapping (Banach-Schauder) theorem, its corollary the inverse mapping theorem, the uniform boundedness principle (Banach-Steinhaus theorem), and the closed graph theorem. In discussing these latter results I will also speak a bit about a result involving countably subadditive seminorms known as Zabreiko’s Lemma (which at its heart is essentially a distillation of Baire Category) that can be used to give alternative proofs of some of these results (e.g. the open mapping theorem). I should also mention the interesting “gliding hump” technique, which provides an alternative proof of Banach-Steinhaus and actually is not just a cloaked version of Baire Category.