$\ell_p$ not a normed subspace of $\ell_q$?

Question

The author defines a normed subspace of a normed linear space as :

A normed subspace of a normed space X is a vector subspace Y of X with the norm obtained by restricting the norm on X to Y.

He later says

$\ell_p$ is not a normed subspace of any $\ell_q$ $(q\neq p)$.

Isn't $\ell_p$ a subset of $\ell_q$ whenever $p<q$? Being a vector space itself, it becomes a vector subspace. And by restricting the norm of the bigger space, can we not say that it is a normed subspace as well?

Answer 1

By "with the norm obtained by restricting the norm on $X$ to $Y$." the author means that for the norm $\Vert \cdot \Vert_Y$ of $Y$ we have $\Vert \cdot \Vert_Y = \Vert \cdot \Vert_X \vert_Y$ i.e. the norm $\Vert \cdot \Vert_X$ restricted onto a subset of its orginal domain.

Indeed we have $\ell_p \subseteq \ell_q$ when $p \leq q$ as vector space, but not as normed spaces.

Explicitely: Even with $p \leq q$ we have $(\ell_p, \Vert \cdot \Vert_p) \nsubseteq (\ell_q, \Vert \cdot \Vert_q)$ since $\Vert \cdot \Vert_p$ and $\Vert \cdot \Vert_q$ do not agree on $\ell_p$. Consider $(1,1, 0, \ldots)$.