X is a infinite dimensional,separable, Banach space with a Hamel Basis $\{e_a\}$. Let $\vert\!\vert x\vert\!\vert=\sum\vert x_a\vert$ , where $x=\sum(x_a e_a)$. Prove that $\vert\!\vert\cdot\vert\!\vert$ is not equivalent to the original norm on X.
I am very confused about where to approach this from. I've tried assuming they are equivalent and proving it wrong to no avail. The other way I have tried to do it is to prove that it will not be complete with the new norm, but i cannot seem to do it that way either. Im not looking for an answer, just a nudge in the right direction.
Thank you <3
Suppose that $||x||_1 =\sum |x_a |$ and $||\cdot ||$ are equivalent. Then $(X, ||\cdot ||_1)$ is a Banach space. But the sequence $$z_n =\sum_{k=1}^n \frac{1}{k^2 } e_{a_k}$$ is an Cauchy sequence in $(X, ||\cdot ||_1)$ without a limit. That gives a contradiction.