Is it true that $p

Question

Let $1 \leq p \leq \infty$ and $1 \leq q \leq \infty$ be two intgers

Is it true that $p<q \Longrightarrow \ell^p \subset \ell^q$

Thanks

I think you mean $\ell^q\subset\ell^p$ – Jason Born Jun 12 '17 at 17:15 — Jason Born, Jun 12 '17 at 17:15
The answer is: yes, it is true. – Jason Born Jun 12 '17 at 17:16 — Jason Born, Jun 12 '17 at 17:16
duplicate of (https://math.stackexchange.com/q/4094). – Jean Marie Jun 12 '17 at 19:11 — Jean Marie, Jun 12 '17 at 19:11

score 3 · Accepted Answer · answered Jun 12 '17 at 17:17

3

Yes. To see this note that if $\sum |a_n|^p < \infty$ then it must be that $|a_n|^p< 1$ for sufficiently large $n$. Hence since $\frac{q}{p} >1$ it follows that $|a_n|^q<|a_n|^p$, which implies the result

answered Jun 12 '17 at 17:17

siegehalver

1,286

1

thanks @Christopher Halverson for your answer – Matey Math Jun 12 '17 at 17:20

Is it true that $p

1 Answers1