Bounded Linear Maps between Complex Normed Spaces and the Operator Norm

Let $\mathbb{R}$ be the ordered field of real numbers, with the notation of that item, contained in the field $\mathbb{C}$ of complex numbers, and let $V$ and $W$ be complex normed spaces with norms $\lVert\cdot\rVert_{V}$ and $\lVert\cdot\rVert_{W}$.

1. (Bounded linear maps)¶ A bound for a linear map $T:V\to W$ is a real number $C$ with $0\le C$ and $\lVert Tv\rVert_{W}\le C\,\lVert v\rVert_{V}$ for every $v\in V$, and $T$ is bounded if it has a bound. We write $\mathcal{L}(V,W)$ for the set of bounded linear maps from $V$ to $W$, and $\mathcal{L}(V)$ for $\mathcal{L}(V,V)$.

2. (Operator norm)¶ Let $T\in\mathcal{L}(V,W)$ and let $B_{T}$ be the set of bounds for $T$. The set $B_{T}$ is nonempty because $T$ is bounded, and it is bounded below by $0$, so it has a greatest lower bound by Existence of the Infimum of a Nonempty Subset of $\mathbb{R}$ Bounded Below. The operator norm of $T$ is