Question about $\ell^{p}$ spaces

Question

I am quite new to the subject of sequence spaces. I got a few doubts (hope they are not silly). While reading about $\ell^{p}$ spaces, I read that these spaces equipt with the $p$-norm form normed linear spaces. My question is every time I am reading about them they are always treated with $p$-norm. Aren't there other norms with which they form normed linear space? For example is it possible that $\ell ^{p}$ space with $(p+k)$-norm (for $k$ greater than $0$) form normed linear space? Also I read that for $0 < p < 1$ $\ell^{p}$ spaces do not carry a norm. I know that the $p$-norm does not follow the triangles inequality for $0 < p < 1$ but why are there no other norms under which we can study $\ell^{p}$ spaces for $0<p<1$?

Answer 1

As you noticed the first problem with a other norm is that you need this norm to be well defined for any element in $\ell^p$ (thus the requirement $k \geq 0$).

The second problem is that, for a lot of reason, the "best" normed linear spaces are Banach spaces. But the (for $k>0$) the space $(\ell^p,\|\cdot\|_{p+k})$, even if it is well defined, is not complete. More over if you take it completion you obtain (for $p+k<+\infty$) the space $\ell^{p+k}$.

(One way to see it is that, for $p<+\infty$, the space $\ell_c$ of sequences with compact support are dense in $(\ell^p,\|\cdot\|_p)$ so $\ell^{p+k}=\overline{\ell_{c}}^{p+k}\overline{\ell^p}^{p+k} \subset \overline{\ell^{p+k}}^{p+k}=\ell^{p+k}$).