The Operator Norm on the Set of Bounded Linear Functionals

# The Operator Norm on the Set of Bounded Linear Functionals

Definition: Let $(X, \| \cdot \|)$ be a normed linear space and let $T$ be a bounded linear operator. The Operator Norm of $T$ is defined to be $\| T \|_{\mathrm{op}} = \inf \{ M : |T(x)| \leq M \| x \|, \: \forall x \in X \}$. |

Later on The Dual Space of a Normed Linear Space page we will see that $X^*$ with the operator norm $\| \cdot \|_{\mathrm{op}}$ is a normed linear space.

The following proposition gives us various ways to define the operator norm of a bounded linear functional.

Proposition 1: Let $(X, \| \cdot \|)$ be a normed linear space and let $T$ be a bounded linear functional. Then:a) $\displaystyle{\| T \|_{\mathrm{op}} = \sup_{x \in X, x \neq 0} \left \{ \frac{|T(x)|}{\| x \|} \right \}}$.b) $\displaystyle{\| T \|_{\mathrm{op}} = \sup_{x \in X, \| x \| = 1} \{ |T(x)| \}}$. |