Let $(V,\|\cdot\|)$ be an infinite dimensional normed space.
Does there alway exist a norm $|||\cdot|||$ on $V$ which induces a strictly coarser topology than $\|\cdot\|$?
I know, that there is always a norm which induces a strictly finer topolgy: We can choose an unbounded linear functional $l:V\to\mathbb K$ and define a new norm as $\|\cdot\|+|l(\cdot)|$. Then
$$\text{Id}:(V,\|\cdot\|+|l(\cdot)|)\to(V,\|\cdot\|)$$
is bounded with unbounded inverse, so $\|\cdot\|+|l(\cdot)|$ induces a strictly finer topolgy. But what about the converse?