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 .

Thus:

We remember that for those two norms to be equivalent in (vector space), there have to be , verifying that: .

Therefore:

But:

So we arrived at a contradiction.