# Functional Analysis, spaces [closed]

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

## closed as off-topic by Saad, dantopa, Shailesh, Cesareo, mrtaurhoApr 29 at 8:57

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• Find a sequence $(a_n)$ so that one of $\sum |a_n|^p$ and $\sum |a_n|^q$ is finite and the other infinite. – David Mitra Apr 18 at 12:36
• @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. – 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?