# The Unapologetic Mathematician

## Mathematics for the interested outsider

I’ve got internet access again, but I’m busy with a few things today, like assembling my furniture. Luckily, Tim Gowers has a post on “How to use Zorn’s lemma”. His example is the construction of additive (not linear) functions from $\mathbb{R}$ to itself.
In practice, as he points out, this is equivalent to defining linear functions (not just additive) from $\mathbb{R}$ to itself, if we’re considering $\mathbb{R}$ as a vector space over the field $\mathbb{Q}$ of rational numbers! This is a nifty little application within the sort of stuff we’ve been doing lately, and it really explains how we used Zorn’s Lemma when we showed that every vector space has a basis.