vector space complete in coarse norm but incomplete in finer norm

Question

I would like to see an example of the following situation.

It is well known that $C([0, 1])$ is complete under the $\|\cdot\|_\infty$ norm, but incomplete under the $\|\cdot \|_1$ norm.

Since clearly in this case, $$ \|\cdot\|_1\leq \|\cdot\|_\infty, $$ I would call $\|\cdot\|_\infty$ the finer norm since it induces a finer metric topology, and $\|\cdot\|_1$ the coarser norm.

I don't think completeness has relationship with the norm being finer or coarser, so I think there exists an example of a vector space $V$ with two inequivalent norms, say $$ \|\cdot\|\leq C\|\cdot\|', \quad\text{for some }C>0, $$ and $V$ is complete in the coarser $\|\cdot\|$, but incomplete in the finer $\|\cdot\|'$.

Thanks in advance for your answer.

Answer 1

Hint: Let $(V,\|.\|)$ be a Banach space and $T: V \to V$ be a linear map whose graph is not closed. [I will let you find an explicit example].

Let $\|x\|'=\|x\|+\|Tx\|$. Then $\|x\|\le \|x\|'$. There is a sequence $(x_n)$ converging to some $x$ such that $(Tx_n)$ converges to some $y \neq Tx$. This sequence $(x_n)$ is Cauchy in $(V,\|.\|')$ but it is not convergent.