nLab operator norm

Idea

(…)

Definition

Definition

For normed vector spaces $\big(V, {\Vert-\Vert_V}\big)$ and $\big(V, {\Vert-\Vert_W}\big)$ the operator norm $\Vert A \Vert$ of a bounded linear operator $A \,\colon\, V \longrightarrow W$ is the supremum of the norms of $A$-images of unit-norm vectors $v \in V$:

(1)$${\Vert A \Vert} \;\coloneqq\; \underset{v \neq 0}{sup} \frac{ {\Vert A(v) \Vert_W} }{ {\Vert v \Vert_V} } \;=\; \underset{ {\Vert v \Vert_V} \leq 1 }{sup} {\Vert A(v) \Vert_W} \,.$$

(cf. Murphy 1990 Ex. 1.1.7)

Remark

If $V$ is not the zero vector space, then the supremum in (1) may equivalently be taken over the unit sphere ${\Vert v \Vert_V} = 1$, instead of the whole unit ball ${\Vert v \Vert_V} \leq 1$, but the latter makes sense also for the degenerate case that $V = 0$ which has no unit vectors.

Related concepts

• norm topology

• operator algebra

• operator spectrum

References

In monographs on operator algebra:

• Gerard Murphy: $C^\ast$-algebras and Operator Theory, Academic Press (1990) [doi:10.1016/C2009-0-22289-6]

See also:

• Wikipedia, Operator norm