From square norm to norm in an inequality

Question

If I have an inequality involving the norm squared of kind: $$||\text{something}||^2<||\phi||^2$$ then can I say that then $$||\text{something}||<a_1||\phi||$$ with $0<a_1<1$?

Answer 1

What you're saying is right. This is because the first inequality implies that $$||s||<||\phi||$$ Therefore, $a_1=\dfrac{1+\dfrac{||s||}{||\phi||}}2$ will do the desired job.