I always recall this anecdote while I was starting getting into Functional Analysis. Vanessa (great student, friend, and TA) always had this strong geometrical intuition. So when we were introduced to Riesz’s Lemma, she always highlighted its strange nature. It implies the closed unit ball of a normed vector space (NVS) is not compact, despite being closed and bounded. She used to do this gesture with her hand à la right-hand rule for showing us that the set is small.

It took me a while to learn to visualize this in a different way. The first part of my exposition is to present different visualizations, where the true smallness of the set is evident.

I deviated from this topic. This conversation sparked my devotion for Functional Analysis, and its techniques.

Small sets in small settings…

The modern definition of compact set is due to Alexandrof and Urysohn. Nevertheless, there were great mathematicians working towards unraveling the meaning of compacness. Big names involve Borel, Heine, Weierstrass, Bolzano…

The main idea is the following: compact sets are equivalent to sequentially compact set, and they are the same as closed and bounded sets. We know this will break on infinite dimensions.

Small sets in big settings… and tight distances!

One cool thing about Banach spaces is their embeddability into spaces of continuous functions. Given (X) a Banach space, consider \(B_{X^}\) to be the the unit ball of the dual of \(X\). Then \(\Phi: X \to C(B_{X^}\), where \(\Phi(x)(x^) = x^(x)\).

We can do better if \(X^\) is separable. We can provide a metric to \(B_{X^})).

I will continue this post in a next post…