Operator norm

Definition

Let $V$ and $W$ be normed spaces and let $T \colon V \to W$ be a continuous linear map. The operator norm of $T$ is $$\|T\| = \sup \{ \|Tx\|_W : x \in V, \ \|x\|_V \le 1 \}.$$

Remarks

The supremum is finite because $T$ is bounded. If $V \neq \{0\}$, then $\|T\| = \sup_{x \neq 0} \|Tx\|_W / \|x\|_V$. It is the smallest $C \ge 0$ with $\|Tx\|_W \le C \|x\|_V$ for all $x \in V$. The operator norm is a norm on the space $L(V, W)$ of continuous linear maps. It satisfies $\|ST\| \le \|S\| \, \|T\|$. If $W$ is a Banach space, so is $L(V, W)$. For a diagonal matrix on $\mathbb{R}^n$ with the Euclidean norm, the operator norm is the largest absolute value of a diagonal entry.

Explanations

L1 human

The operator norm tells you how much $T$ can stretch a vector at most. For example, the map $x \mapsto 3x$ on $\mathbb{R}$ has operator norm $3$.