A week ago, we were asked in Fourier Series class to find a counterexample to prove that norms aren’t equivalent in vector space .
The definition for this space is:
The norm based on the dot product is:
We can define another norm as: . It’s easy to prove that it’s a norm.
Let , with for all .
We remember that for those two norms to be equivalent in (vector space), there have to be , verifying that: .
So we arrived at a contradiction.