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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 $0<p<1$?

Errol
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ogirkar
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  • What exactly do you mean by $(p+k)$ norm? Are you asking whether there is a $k>0$ such that if $f\in\ell^p$, then $f\in \ell^{p+k}$? – Alex R. Sep 25 '18 at 20:10
  • Means I am asking that whether $(l^{2},3-norm) $ will form normed linear space or not? – ogirkar Sep 26 '18 at 13:40

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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 $\ell^{p+k}=\overline{\ell_{c}}^{p+k}\overline{\ell^p}^{p+k} \subset \overline{\ell^{p+k}}^{p+k}=\ell^{p+k}$).

Delta-u
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  • Ok.Means $(l^{2},3-norm)$ is normed linear space but it is not Banach space.And why $l^{p}$ spaces does not carry a norm for $0<p<1$? – ogirkar Sep 26 '18 at 13:49
  • Indeed :-). For $0<p<1$ you can associate a norm (for example a $q$-nom, $q \geq 1$)but it will not be a Banach space. It can me a complete space with the metric associated with the "$p$-norm" but as this is not locally convex you can't have a norm taht make $l^p$ a Banach space. – Delta-u Sep 26 '18 at 13:56
  • Thanks.just one question.How metric associated with p-norm is given.1)$d(x,y)=\sum |x_{i}-y_{i}|^{p}$ or 2)$(\sum (|x_{i}-y_{i}|^{p}))^{1/p}$.I think first formula will hold for $p>0$ and second only for $p\geq 1$ – ogirkar Sep 26 '18 at 16:48
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    I think exactly the same :-). – Delta-u Sep 26 '18 at 16:51