Proof of supremum norm being norm on vectorroom

[Luisa]

Hey, I have quite a problem rn and don’t really know how to start at all:

Let [a,b] be a finite interval. B([a,b],C) (C being the sign for complex numbers) be the room of the stetic function [a,b]->C. Show that ||f||_inf:= sup_{x element of [a,b]} |f(x)| is a norm on B([a,b],C).

I already proofed that B is a vector room, my prof told me to do so, but idk how to continue now.. Thanks for trying to help!

 

[Dr.Peterson]

Hey, I have quite a problem rn and don’t really know how to start at all:

Let [a,b] be a finite interval. B([a,b],C) (C being the sign for complex numbers) be the room of the stetic function [a,b]->C. Show that ||f||_inf:= sup_{x element of [a,b]} |f(x)| is a norm on B([a,b],C).

I already proofed that B is a vector room, my prof told me to do so, but idk how to continue now.. Thanks for trying to help!

First, you should know that in English, we say "vector space", not "vector room". I understand that in many languages the two words are the same!

What you need to do now is to look up the definition of a norm, in your textbook or elsewhere, and then write out how each property of a norm applies to this function. It may be almost obvious, once you've done this, that they are all true, or it may take some thought. Please show your work as far as you get.