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 $f$ be a bounded linear functional on $X$. The Operator Norm of $f$ is defined to be $\| f \|_{\mathrm{op}} = \inf \{ M : |f(x)| \leq M \| x \|, \: \forall x \in X \}$. |

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

Proposition 1: Let $(X, \| \cdot \|)$ be a normed linear space and let $f$ be a linear functional on $X$. If there exists an $M > 0$ such that $|f(x)| \leq M$ for every $x \in X$ then $f$ is bounded. |

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

Proposition 2: Let $(X, \| \cdot \|)$ be a normed linear space and let $f$ be a bounded linear functional. Then:a) $\displaystyle{\| f \|_{\mathrm{op}} = \sup_{x \in X, x \neq 0} \left \{ \frac{|f(x)|}{\| x \|} \right \}}$.b) $\displaystyle{\| f \|_{\mathrm{op}} = \sup_{x \in X, \| x \| = 1} \{ |f(x)| \}}$. |