If p $\neq$ q, Show that it implies $\ell_p$ $\neq$ $\ell_q$ $\\$

I am new to functional Analysis, I don't know how to go about this.


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  • $\begingroup$ Find a sequence $(a_n)$ so that one of $\sum |a_n|^p$ and $\sum |a_n|^q$ is finite and the other infinite. $\endgroup$ – David Mitra Apr 18 at 12:36
  • $\begingroup$ @DavidMitra Please can you give me an example of one. I have racked my brains, since after you left the comment above. But I couldn't find anything, no headway. Thanks. $\endgroup$ – Vincent Ebuka Apr 23 at 20:55

Here is another hint: $\ell_1 \not= \ell_2$ because $$ \sum_{k=1}^\infty \frac 1k = \infty \quad \text{and} \quad \sum_{k=1}^\infty \frac 1{k^2} < \infty.$$ Can you modify this example?

  • $\begingroup$ Not really for now, but I will study it closely. Thanks for the hint. $\endgroup$ – Vincent Ebuka Apr 18 at 12:59
  • $\begingroup$ I have tried but couldn't. Any further hint/help will be appreciated.Thanks. $\endgroup$ – Vincent Ebuka Apr 23 at 21:05

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