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Norms question (parallelogram law)

  • Thread starter 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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Answers and Replies

  • #2
tiny-tim
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hi norm! :biggrin:

i don't understand what you're doing :confused:

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

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