Operator norm - Knowlpedia

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

\|T\|=\sup_{\|v\|_V\le 1}\|T(v)\|_W,

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

Equivalent formulas

For $V\ne\{0\}$ , this also equals

\sup_{v\ne0}\frac{\|T(v)\|_W}{\|v\|_V}=\sup_{\|v\|_V=1}\|T(v)\|_W.

For the zero vector space, the unit-ball formula gives $\|T\|=0$ .

Remarks

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 $\|A\|=\max_i |d_i|$ .