I'm very new to the practice of computing operator norm and for every question I seem to get stuck unless it is trivially easy (or probability even when it is still trivially easy). Basically I cannot seem to be able to compute even the simplest operator norm. For example, consider the linear operator
$$ T: \ell^1 \rightarrow \ell^{\infty}: (x_k)_{k=1}^{\infty} \mapsto (x_k/2^{k})_{k=1}^{\infty}.$$ In order to compute the operator norm I can see that $$||Tx||_{\ell^{\infty}} = \sup_{k}|x_k/2^k| \leq \sum_{k=1}^{\infty}|x_k|/2^k \leq \sum_{k=1}^{\infty}|x_k|=||x||_{\ell^1}.$$ Then by taking sup over all $ \||x||_{\ell^1} = 1 $ we have $||T||\leq 1$. So a good guess seems to be that $||T||=1$. But I cannot prove the other inequality. For right now the most reasonable approach I can think of is to cook up a sequence such that $ ||x||_{\ell^1} \leq 1 $ and $||Tx||_{\ell^\infty} = 1$. Can anyone help me with this problem?
In fact, after going through some questions (without solving anyof them) it seem I can always easily obtain some upper bound of $||T||$ but then get stuck on the other inequality as soon as the function $T$ becomes reasonably complex (or not). Is this even the right mind set when attacking such a problem? I would like some general advice if anyone could be kind enough to offer them to me.
Thank you!