# John's theorem and factorizations of linear operators

## The correspondence between convex sets and finite dimensional Banach spaces

Let $K \subseteq \R^n$ be a symmetric compact convex set with nonempty interior. Such a convex set is called a convex body. We can associate a norm with $K$, called the gauge of $K$, as follows:

$\|x\|_K = \inf \{t \geq 0 \mid x \in t K\}$

It readily follows that $\| \cdot \|_K$ is the norm such that $K$ is the unit ball of that norm. A finite dimensional Banach space is essentially $\R^n$ using some $K$ as the gauge of a norm. Further, the unit ball of any Banach space is a convex body; thus symmetric convex sets and finite dimensional Banach spaces are practically interchangeable. The prototypical example is $\ell_2^n$, whose unit ball is the regular 'spherical' ball $\{x \in \R^n \mid \sum_{i=1}^n x_i^2 \leq 1\}$. This correspondence lets us translate inequalities with applications to functional analysis to nice (at least in my opinion) geometric statements about the relations of different convex bodies to each other.

## Factorizations

Theorem.(Dvoretzky-Rogers Lemma)

Let $(X, \|\cdot\|_X)$ be an $n$-dimensional Banach space. Then there exists a set of $m = \sqrt{n}$ vectors $x_1, \ldots, x_m \in \R^n$ such that for all $t_1, \ldots, t_m \in \R$

$\max_{1 \leq i \leq m} |t_i| \leq \left\| \sum_{i=1}^m t_i x_i \right\|_X \leq C \cdot \left(\sum_{i=1}^m t_i^2 \right)^{1/2}$

where $C > 0$ is a universal constant.