Define the forward shift map TinL$(F^{})$ by

$T(x_1,x_2,x_3,$dots$)=(0,x_1,x_2,x_3,$dots$).$

Prove that $T$ is injective, but not surjective. $($While you do not need to show that $T$ is linear, it would be a good exercise to work out the details in your own time.$)$