# Geoemtric interpretation of the norm of a bounded linear functional

• March 28th 2011, 11:38 PM
raed
Geoemtric interpretation of the norm of a bounded linear functional
Dear Colleagues,

Show that the norm $||f||$ of a bounded linear functional $f\neq 0$ on a normed space $X$ can be interpreted geometrically as the reciprocal of the distance $d=inf\{||x|| \ |f(x)=1\}$ of the hyperplane
$H_{1}=\{x\in X \ |f(x)=1\}$ from the origin.

Regards,

Raed.
• April 7th 2011, 06:19 PM
mr fantastic
Quote:

Originally Posted by raed
Dear Colleagues,

Show that the norm $||f||$ of a bounded linear functional $f\neq 0$ on a normed space $X$ can be interpreted geometrically as the reciprocal of the distance $d=inf\{||x|| \ |f(x)=1\}$ of the hyperplane
$H_{1}=\{x\in X \ |f(x)=1\}$ from the origin.