Past exam of the analysis course of the University of Oxford / 2010 / A / 1 / d / Solution

For the first linear map, take $(\alpha ,\beta )=(0,e_n)$. Its input norm is one, while $T_1(0,e_n)=(e_n,0)$ has norm $n^2$. Thus $T_1$ is unbounded.

For the second map, since $j^2\ge 1$,Equality holds at $(0,e_1)$, soBoth formulas map $c_{00}\times c_{00}$ into itself because adding two finitely supported sequences preserves finite support.