Answer to Question #109043 in Functional Analysis for ahmed alin

Question #109043

prove that

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


1
Expert's answer
2020-04-16T18:41:37-0400

"||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"


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