# Functions not satisfying parallelogram identity with supremum norm

1. Mar 3, 2010

1. The problem statement, all variables and given/known data
Find two functions $$f, g \in C[0,1]$$ (i.e. continuous functions on [0,1]) which do not satisfy

$$2 ||f||^2_{sup} + 2 ||g||^2_{sup} = ||f+g||^2_{sup} + ||f-g||^2_{sup}$$

(where $$|| \cdot ||_{sup}$$ is the supremum or infinity norm)

2. Relevant equations
Parallelogram identity: $$2||x||^2 + 2||y||^2 = ||x+y||^2 + ||x-y||^2$$ holds for any x,y

3. The attempt at a solution
Honestly no idea.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Mar 3, 2010

### Dick

Just try some functions. It's really not hard to find an example that doesn't work.

3. Mar 3, 2010