Norms question (parallelogram law)

[tourjete]

Homework Statement

Consider the vector space C[a,b] of all continuous complex-valued functions f(x), x\in [a,b]. Define a norm ||f|| = max{|f(x)|, x\in [a,b]]

a) show that this is a norm
b) Show that this norm does not satisfy the parallelogram law, thereby showing that its not an inner-product norm.

Homework Equations

Parallelogram law: ||x-y||^{2} + ||x+y||^{2} = 2||x||^{2} + 2||y||^{2}

The Attempt at a Solution

I'm mostly having trouble defining the norm. I'm a little unclear on what the concept of a norm is; we only went over inner-product norms in class. I draw the vector going from the origin to the maximum on [a,b] and to define the norm I wrote ||f|| = \sqrt{(([vcos])^2 + ([vsin])^2} since that would make it always positive. When I used the parallelogram law, I used x = vcos(theta) and y = vsin(theta). However, I clearly defined the dorm wrong since I got that the two sides of the equation equaled each other.

Is there another way to define a norm? Am I choosing x and y in the parallelogram law wrong?

 

[tiny-tim]

hi norm!

i don't understand what you're doing …

if eg f(x) = x, g(x) = -x, then ||f+g|| = 0, but ||f|| = ||g|| = b