- #1

tourjete

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## Homework Statement

Consider the vector space C[a,b] of all continuous complex-valued functions f(x), x[tex]\in[/tex] [a,b]. Define a norm ||f|| = max{|f(x)|, x[tex]\in[/tex] [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||[tex]^{2}[/tex] + ||x+y||[tex]^{2}[/tex] = 2||x||[tex]^{2}[/tex] + 2||y||[tex]^{2}[/tex]

## 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|| = [tex]\sqrt{(([vcos])^2 + ([vsin])^2}[/tex] 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?

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