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Functions not satisfying parallelogram identity with supremum norm

  • Thread starter JackTheLad
  • Start date
  • #1

Homework Statement


Find two functions [tex]f, g \in C[0,1][/tex] (i.e. continuous functions on [0,1]) which do not satisfy

[tex]2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup}[/tex]

(where [tex]|| \cdot ||_{sup}[/tex] is the supremum or infinity norm)

Homework Equations


Parallelogram identity: [tex]2||x||^2 + 2||y||^2 = ||x+y||^2 + ||x-y||^2[/tex] holds for any x,y


The Attempt at a Solution


Honestly no idea.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
26,258
618
Just try some functions. It's really not hard to find an example that doesn't work.
 
  • #3
For posterity, two functions which fit nicely are
f(x) = x
g(x) = x-1


(I had tried lots of functions but they worked; not very helpful response)
 

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