Operator norm

An operator norm of a linear map $T:V\to W$ between normed vector spaces $(V,\|\cdot\|_V)$ and $(W,\|\cdot\|_W)$ is the quantity

with the understanding that the supremum may be $+\infty$ in general. When $\|T\|<\infty$, one says $T$ is bounded.

For linear maps between normed spaces, finiteness of the operator norm is equivalent to being continuous . The operator norm makes the collection of bounded linear maps into a normed vector space and specializes to a norm on linear operators when $V=W$.

Examples:

• If $T(v)=c\,v$ on a normed space, then $\|T\|=|c|$.
• For the projection $P(x,y)=(x,0)$ on $\mathbb{R}^2$ with the Euclidean norm, $\|P\|=1$.
• For a diagonal matrix $A=\operatorname{diag}(d_1,\dots,d_n)$ acting on $\mathbb{R}^n$ with the max norm, the induced operator norm is $\|A\|=\max_i |d_i|$.