Answer to Question #109043 in Functional Analysis for ahmed alin

Question

prove that

||u||>0 for u<>0

Answer 1

$||u||=\sqrt{(u,u)}$

For scalar u;

$(u,u)=u^2$

For vector u;

$(u,u)=u.u$

For complex u;

If $u=(u_1,u_2,...,u_n)$

$(u,u)=u_i\bar{u_i}$

For scalar u if $u<>0$ ;

Case I $u<0$

$u=-x$

$||u||=\sqrt{(-x)(-x)}=\sqrt{x^2}=x>0$

Case II $u>0$

$u=x$

$||u||=\sqrt{x.x}=\sqrt{x^2}=x>0$

So,$||u||>0$