Let $L:V \to W$ be a linear map between two normed vector space. Let $E \subseteq V$ be a compact subspace, then we know that $L$ takes on a maximum and a minimum on $E$, since $L$ is continuous and $E$ compact.
My question: what is the definition of a maximum and a minimum in normed vector spaces? As far as I know it means that the norm is maximal/minimal, so for the maximum $\vec{x}$ we have $\|f(\vec{x})\| \geq \|f(\vec{y})\|$ for all $\vec{y} \in E$, and similarly for minimum.
However, if I take this further to $\mathbb{R}$, where the norm is the absolute value $|x|$, how does this make sense? That would mean that a negative absolute minimum could become an absolute maximum if we take absolute value.
The result that a continuous map from a compact subset of a topological space to $\mathbb{R}$ achieves its minimum and maximum values always holds, whether or not the map arises from taking the norm of a continuous map as in your question. Typically, when one talks about finding extrema of a map the range space is $\mathbb{R}$. When we talk about the min or max of a (continuous) linear map between normed vector spaces, it is implicitly assumed that we are talking about the min or max of the map $||L|| : V \to \mathbb{R}$.