Operator norm of a functional

Question

So I know the definition of a operator norm of a matrix. Suppose a functional takes a vector space to a scalar, what is the definition of operator norm of a functional then? Doesn't really make sense to me, unless there are different definitions of operator norms. Can someone clarify?

Answer 1

In general, if $V$ and $W$ are normed linear spaces, and $U: V \to W$ is a linear operator, we define the norm of $U$ as

$$\| U \| = \inf \{M>0 \mid \| U(x) \| \le M \| x \| \ \forall x \in V \} = \sup _{\| x \| \le 1} \frac {\| U(x) \|} {\| x \|} = \sup _{\| x \| = 1} \frac {\| U(x) \|} {\| x \|}$$

(the theree values are equal).

In particular, if $W = \Bbb C$ (or $\Bbb R$), just replace the norm in $W$ with the usual absolute value.

Answer 2

The operator norm always compares the size of the image vector to the size of the input vector, and you measure size using whatever the relevant norm is on either space. So, the general definition of an (induced) operator norm for a mapping $A:X\rightarrow Y$ where $X$ and $Y$ are normed vector spaces is

$$ \|A\|=\sup_{x\neq 0}\frac{\|Ax\|_Y}{\|x\|_X} $$ So, what about linear functionals? In this case, $Y = \Bbb{R}$ or $Y = \Bbb{C}$, so $\|Ax\|_Y = \vert Ax\vert$, the absolute value, and so if $T:X\rightarrow \Bbb{R}$ (or $\Bbb{C}$) is a linear functional,

$$ \|T\| = \sup_{x\neq 0}\frac{\vert Tx\vert}{\|x\|_X} $$