Let $\displaystyle s:l^2\to l^2$ by the map

$\displaystyle s(x_0,x_1,\cdots )=(0,x_0,\cdots )$

Is it linear?

Let $\displaystyle x_i,y_i\in l \ \text{and} \ \lambda,\mu\in F$

$\displaystyle s(\lambda x_i +\mu y_i)=s(\lambda x_0 + \mu y_0, \lambda x_1 +\mu y_1,\cdots )$$\displaystyle =(0,\lambda x_0 + \mu y_0, \lambda x_1 +\mu y_1,\cdots )$

$\displaystyle =(0,\lambda x_0,\cdots )+(0,\mu y_0,\cdots)=\lambda(0,x_0,\cdots)+\mu(0,y_0,\cdots )$$\displaystyle =\lambda s(x_i)+\mu s(y_i)$

Yes, the map is linear.

Is it monic?

If $\displaystyle s(x_i)=s(y_i)$, then $\displaystyle (0,x_0,\cdots)=(0,y_0,\cdots)$

$\displaystyle (0,x_0,\cdots)-(0,y_0,\cdots)=(0,x_0-y_0,\cdots)=0$

$\displaystyle \Rightarrow x_i-y_i=0\Rightarrow x_i=y_i$

Yes, the map is monic.

How do I show it is an epimorphism?