Showing that one cannot continuously embed $\ell^\infty$ in $\ell^1$.

Question

Is it possible to embed $\ell^\infty$ into $\ell^1$ continuously? I.e. can one find a continuous linear injection $I:\ell^\infty \to \ell^1$.

I have reduced a problem I have been working on to showing that this cannot happen, but I don't see how to proceed from here.

Answer 1

Yes, it's possible; for example, you can set $$ I(a_1,a_2,a_3,\dots):=(\frac{a_1}{1^2},\frac{a_2}{2^2},\frac{a_3}{3^2},\dots). $$