I'm studying Introductory Functional Analysis with Applications by Erwin Kreyszig. In Chapter 2, section 2.4, while solving exercise, this question came to my mind, but I can't figure it out, please help me...
If $\ell^p$ is metric space, where $1≤p<∞$, then show that whether $\ell^{p-1}$ is subspace of $\ell^p$ or not? If it is not a subspace, give reason.
For $p=1$, this doesnt really make sense unless you want to be bold and tell the world what $0^0$ is.
In general, let $0<s<t<\infty$ and let $(x_n)\in\ell^s$. Then $\sum_n|x_n|^s<\infty$. In particular $|x_n|^s\to0$ and thus we have some $n_0$ such that $|x_n|<1$ for $n\ge n_0$. Now since $s<t$, we have that $$|x_n|^t=|x_n|^s|x_n|^{t-s}=|x_n|^s\cdot\text{ number less than 1}\le |x_n|^s$$ for all $n\ge n_0$. thus $$\sum_n|x_n|^t=\sum_{n=1}^{n_0}|x_n|^t+\sum_{n>n_0}|x_n|^t\le \text{finite number}+\sum_{n>n_0}|x_n|^s<\infty$$ so $(x_n)\in\ell^t$, i.e. $\ell^s\subset\ell^t$.
For $1< p<\infty$, take $t=p$ and $s=p-1$.