Discontinuity of a Linear Map

Question

Consider the Banach spaces

$l^1(N)=\{\xi:N\rightarrow C:||\xi||_1=\sum_{n=1}^\infty |\xi(n)|<\infty \}$

$l^2(N)=\{\xi:N\rightarrow C:||\xi||_2=(\sum_{n=1}^\infty |\xi(n)|^2)^{1/2}<\infty \}$

Is the linear map $J:l^1(N)\rightarrow l^2(N)$, defined by $J(\xi)=\xi$ not continuous? Why?

Answer 1

$\sum |\xi_n|^{2} \leq (\sum |\xi_n|)^{2}$ so $J$ is continuous and $\|J\| \leq1 $