Consider a mapping $T_\lambda: \ell^1 \rightarrow \ell^1\quad T_\lambda f:=\{\lambda_1 f_1,\,\lambda_2 f_2,\lambda_3 f_3,\,\cdots\},$ where $\lambda_n = 1 - \frac{1}{n}$, $\lambda \in \ell^\infty$.
The operator norm is not attained, which can be shown by Hölder inequality. But I am looking for an alternative proof that is more elementary and avoids from using 'advanced' theories such as Hölder inequality.
Any suggestions?
Thanks.