Operator norm

Let $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ be normed vector spaces over $\mathbb{F}$, and let $T:V\to W$ be linear. The operator norm (or induced norm) of $T$ is

Equivalently,

The operator norm measures the largest factor by which $T$ can stretch vectors. It is fundamental in analysis because boundedness/continuity of linear maps between normed spaces is equivalently expressed by finiteness of $\|T\|$.

Examples:

• If $T:\mathbb{R}\to\mathbb{R}$ is $T(x)=ax$, then $\|T\|=|a|$ (with the usual absolute value norm).
• If $T:\mathbb{R}^k\to\mathbb{R}^k$ is the identity map, then $\|T\|=1$.
• If $T:\mathbb{R}^2\to\mathbb{R}^2$ is rotation by angle $\theta$, then $\|T\|=1$ (it preserves Euclidean length).