Are $\ell_p$ spaces complete under the $q$-norm? $\ell_p$ spaces are complete under the $p$-norm. The question is:

if we change the norm, what can we say about the completness of the space?

For example:
$\ell_1$ (the space of all summable sequences)
is complete or not in the two norm? If it's not complete is there a counter example showing that?
Also the same question for $L_p$ spaces
(Lesbegue $p$-integrable functions)  with $q$-norm

In general: no.
In the cases that $\mathscr l^p(X)$ or $L^p(X)$ end up being finite dimensional: yes.
For $p<q$ you have $\mathscr l^p(X)\subset \mathscr l^q(X)$ but $\mathscr l^p(X)\neq\mathscr l^q(X)$ if $X$ is not a finite set. If $\mathscr l^p(X)$ were complete with $\|\cdot\|_q$ norm, then it would have to be a closed subset of $\mathscr l^q(X)$. But the space of finitely supported functions on $X$ $\mathscr l_0(X)$ is dense in $\mathscr l^q$ and lies in $\mathscr l^p$. So:
$$\overline{\mathscr l^p(X)}\supseteq\overline{\mathscr l_0(X)}=\mathscr l^q(X) \supsetneq \mathscr l^p(X)$$
and $\mathscr l^p(X)$ is not closed in $\mathscr l^q(X)$ and so not complete.
As to $L^p$ spaces: Note that depending on the measure space $X$ you can have both:
$$L^p(X)\not\subset L^q(X)\qquad L^p(X)\not\supset L^q(X)$$
whenever $q\neq p$ and the question is ill defined.
But for example if $X$ is a bounded open subset of $\mathbb R^n$ you have $L^p(X)\supset L^q(X)$, $L^p(X)\neq L^q(X)$ when $p<q$ and you can ask if $L^q$ is complete in $\|\cdot\|_p$ norm. The answer is again no with the same reasoning. The continuous functions on $X$ are in this case a dense subset in both $L^p$ and $L^q$ and the same reasoning as before applies.
