A week ago, we were asked in Fourier Series class to find a counterexample to prove that norms aren’t equivalent in vector space $L^2(a,b)$.

The definition for this space is:

The norm based on the dot product $% = \int_a^b f \cdot g %]]>$ is:

We can define another norm as: $\|f\|_{*} = \int_a^b \lvert f \rvert$. It’s easy to prove that it’s a norm.

Let $f_n: (a,b) \rightarrow \mathbb{R}$, with $f_n(x)= e^{nx}$ for all $n\in \mathbb{N}$.

Thus:

We remember that for those two norms to be equivalent in $L^2(a,b)$ (vector space), there have to be $\lambda, \delta \in \mathbb{R^{+}}$, verifying that: $\delta \|f\|_{*} \le \|f\|_2 \le \lambda \|f\|_{*} \quad \forall f\in L^2(a,b)$.

Therefore:

But:

So we arrived at a contradiction.