Answer to Question #308219 in Functional Analysis for mathforfun

Question

prove ||Ax|| < ||A|| ||x||

Answer 1

Solution

Let us suppose that

$||Ax|| > ||A|| ||x||$

$\Rightarrow\frac{||Ax||}{||x||} > ||A||$

$\Rightarrow|A\frac{x}{||x||}| > ||A||$

Since $\frac{x}{||x||}|$ is a unit vector, therefore, from above we can write,

$\Rightarrow ||A|| > ||A||$ which constradicts

Hence

$||Ax|| < ||A|| ||x||$

Proved