I'm working through a book on differential equations, and I'm at the point where the author is justifying the use of the matrix exponential. So, given $T: \mathbb{R}^n \to \mathbb{R}^n$ a linear operator, he defines the norm $\| T \| = max_{|x| \le 1} |T(x)|$ where $|x|$ is the standard Euclidean norm. I'm confused how he can define this by using the max instead of the sup. On the wikipedia page for matrix norms, there's discussion about how one can derive an induced norm, where this induced norm is written with respect to the maximum function.
Since we're working over an uncountable set (the unit ball), how do we know a maximum even exists? Can somebody elucidate this?
A linear transformation $T: \mathbb{R^n} \to \mathbb{R}^n$ is continuous (easily verified), and the euclidean norm $|\cdot|$ is also continuous so, the composite map $ |\cdot| \circ T$, given by $x \mapsto |T(x)|$ is continuous. Also, the unit ball in $\mathbb{R}^n$ is compact (Heine-Borel). Hence, by the extreme value theorem, the maximum is indeed attained (and is finite) hence it equals the $\sup$. In fact there is nothing special about $\mathbb{R}^n$; you can replace it with any finite dimensional vector space $V$ over $\mathbb{R}$, but finite-dimensionality is crucial.
In general, if the spaces are infinite dimensional, the unit ball need not be compact, so the maximum need not be attained, then we have to replace $\max$ by $\sup$.