Definition (Operator norm)

For some linear map $A\in \mathscr{L}(\mathbb{R}^{n};\mathbb{R}^{m})$, the expression $$\lVert A \rVert _{\mathbb{R}^{n},\mathbb{R}^{m}}=\sup\{ \lVert A(\boldsymbol x) \rVert _{\mathbb{R}^{m}}\mid \lVert \boldsymbol x \rVert _{\mathbb{R}^{n}}=1 \}$$or for a linear functional $f$ $$\lVert f \rVert = \sup_{x:\lVert x \rVert \neq 0} \frac{|f(x)|}{\lVert x \rVert }$$ is the operator norm of $A$.

Definition (Bounded in operator norm)

Let $X$ be a normed vector space. We say $f\in\Gamma(X;\mathbb{R})$ is bounded (in the operator norm) if $\exists M>0$ such that $$|f(x)|\le M\lVert x \rVert \quad\forall x\in X$$