Operator Norm

Let $T:V\to W$ be a linear map between normed vector spaces. The operator norm of $T$ is

When $V$ is finite-dimensional (for example $V=\mathbb{R}^k$ with the Euclidean norm ), this supremum is finite and is attained by some unit vector.

The operator norm is submultiplicative: $\|S\circ T\|\le \|S\|\,\|T\|$ whenever the composition makes sense.

Examples:

• If $T:\mathbb{R}^k\to\mathbb{R}^k$ is scalar multiplication by $c\in\mathbb{R}$, then $\|T\|=|c|$.
• The orthogonal projection $P:\mathbb{R}^2\to\mathbb{R}^2$, $P(x,y)=(x,0)$, has $\|P\|=1$.
• Any rotation of $\mathbb{R}^2$ has operator norm $1$ (it preserves Euclidean lengths).