Some closed subspace of $l_2$?

Question

$(a)$ I was trying to define a continuous linear map $T$ on $l_2$ whose kernel would be the $A$ and can conclude $A=T^{-1}(0)$ and hence closed set? could anyone help me to solve any of one?

Answer 1

a) $$T(x_1,x_2,x_3,...)=(x_2-2x_1,x_4-4x_3).$$ It is finite rank and hence continuous, or alternatively $||T(x)||\leq\sqrt{17}||x||$.

(b) $$T(x_1,x_2,x_3,...)= (0,x_2,0,x_4,0,...).$$ For which $||T(x)||\leq ||x||$.

(c) $$T(x_1,x_2,x_3,...)= (x_1,0,x_3,0, ...).$$ For which $||T(x)||\leq||x||$.