The Operator Norm of a Real Matrix

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m,n\in\mathbb{N}$ and let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ be a real $m\times n$ matrix, with the matrix-vector product $Ax$ of $A$ and a point $x$ of $\mathbb{R}^{n}$, the Euclidean norm $\lVert\cdot\rVert$ on $\mathbb{R}^{n}$ and on $\mathbb{R}^{m}$, and suprema of subsets of $\mathbb{R}$. Let $U_{n}=\{x\in\mathbb{R}^{n}:\lVert x\rVert\le1\}$, and for every $B\in\mathcal{M}_{m\times n}(\mathbb{R})$ let $N_{B}=\{\lVert Bx\rVert:x\in U_{n}\}$, the set of the values $\lVert Bx\rVert$ at the points $x\in U_{n}$, be as in The Set of Lengths ‖Ax‖ over the Unit Ball Has a Supremum, and the Supremum Map Exists; in particular $N_{A}=\{\lVert Ax\rVert:x\in U_{n}\}$, which has a supremum by The Set of Lengths ‖Ax‖ over the Unit Ball Has a Supremum, and the Supremum Map Exists §supremum.

¶ The operator norm of $A$ is

$$\lVert A\rVert=\sup N_{A},$$

the value at $A$ of the map $\mathcal{M}_{m\times n}(\mathbb{R})\to\mathbb{R}$, $B\mapsto\sup N_{B}$, which exists and is unique by The Set of Lengths ‖Ax‖ over the Unit Ball Has a Supremum, and the Supremum Map Exists §map. Thus $\lVert\cdot\rVert$ is that map from $\mathcal{M}_{m\times n}(\mathbb{R})$ to $\mathbb{R}$.

¶ $\lVert A\rVert\in\mathbb{R}$, as the value at $A$ of the map of The Set of Lengths ‖Ax‖ over the Unit Ball Has a Supremum, and the Supremum Map Exists §map, by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol §graph.

Notation: $\lVert\cdot\rVert$ also denotes the Euclidean norm of a point; the argument shows which is meant.