Let $X$ and $Y$ be normed vector spaces over the same scalar field $\mathbb F$, where $\mathbb F$ is $\mathbb R$ or $\mathbb C$. Write $0_X$ and $0_Y$ for the zero vectors of $X$ and $Y$, respectively. Let $\mathcal L(X,Y)$ denote the [vector space](/page/Vector%20Space) of bounded $\mathbb F$-linear maps $T:X\to Y$, with pointwise addition and scalar multiplication. For $T\in\mathcal L(X,Y)$, define the [operator norm](/page/Operator%20Norm) by

\begin{align*} \|T\|_{\mathcal L(X,Y)}:=\sup\{\|Tx\|_Y:x\in X,\ \|x\|_X\le 1\}. \end{align*}

Then the map

\begin{align*} \mathcal L(X,Y)&\to[0,\infty) \end{align*}

\begin{align*} T&\mapsto \|T\|_{\mathcal L(X,Y)} \end{align*}

is a norm on $\mathcal L(X,Y)$.