Where can I find a proof of the supremum norm as a norm?

[e^ipi=-1]

Could anyone tell me where to find a proof of the fact that the supremumnorm is a norm?

The supremum norm is also known as the uniform, Chebychev or the infinity norm.

 

[radou]

You could start with writing down the general definition of a norm and what it must satisfy.

 

[e^ipi=-1]

I already proved three of the conditions but I am stuck on the triangle equality. Also I wanted to see how other people did it.

 

[radou]

The proof of the triangle inequality is a consequence of the "standard" triangle inequality.

Edit: and of the property of the supremum, that sup{f(x) + g(x), x in S} <= sup{f(x), x in S} + sup{g(x), x in S}, for any functions f, g.

 

[e^ipi=-1]

The property of the supremum is exactly where I got stuck. I had the feeling that the property holds, but I could't figure out why. Could you tell me?

 

[HallsofIvy]

Do you mean "sup(f+ g)\le sup(f)+ sup(g)"? That's pretty close to trivial.

Let x be any value. Then f(x)\le sup(f) and g(x)\le sup(g). Together, f(x)+ g(x)\le sup(f)+ sup(g). That is, sup(f)+ sup(g) is an upper bound on f(x)+ g(x) and since sup(f+ g) is the least upper bound, sup(f+ g)\le sup(f)+ sup(g).

 

[e^ipi=-1]

Thanks a lot. Pretty stupid I didn't think of that.