Functional Analysis, spaces

Question

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.

Answer 1

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?