$(x,Ay):=\sum_{k=1}^{\infty}x_k y_k$ and $A:\ell^1 \to c_0^*$ and $x_k\in c_0, y_k\in\ell^1$
I would like to show A is surjective and isometric.
I am not sure about the deifnition of an isometriy, does it mean I have to show that $(x,y)=(Ax,Ay)$ ?
Since $y\in\ell^1$ I know that $\sum_{k=0}^{\infty}|y_k|<\infty$ and because of $x\in x_0$ I know $\lim x_n=0$ This could help me for an infinite sum.
How can I write $(x,y)$ and $(Ax,Ay)$ now?