Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m,n,p\in\mathbb{N}$, let $A,B\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $C\in\mathcal{M}_{n\times p}(\mathbb{R})$ be real matrices, and let $t,c\in\mathbb{R}$, with the sum $A+B$ and scalar multiple $tA$, the zero matrix $0\in\mathcal{M}_{m\times n}(\mathbb{R})$, the matrix product $AC\in\mathcal{M}_{m\times p}(\mathbb{R})$, the matrix-vector product $Ax$, the Euclidean norm $\lVert x\rVert$ of a point $x$ of $\mathbb{R}^{n}$, the operator norm $\lVert\cdot\rVert$ of matrices, and the absolute value $|t|$.

¶ $0\le\lVert A\rVert$.

¶ $\lVert Ax\rVert\le\lVert A\rVert\,\lVert x\rVert$ for every $x\in\mathbb{R}^{n}$.

¶ If $\lVert Ax\rVert\le c\,\lVert x\rVert$ for every $x\in\mathbb{R}^{n}$, then $\lVert A\rVert\le c$.

¶ $\lVert A\rVert=0$ if and only if $A=0$.

¶ $\lVert tA\rVert=|t|\,\lVert A\rVert$.

¶ $\lVert A+B\rVert\le\lVert A\rVert+\lVert B\rVert$.

¶ $\lVert AC\rVert\le\lVert A\rVert\,\lVert C\rVert$.