Is p(X) closed in X** for a Banach space X?

[math8]

Let X be a Banach space. We show p(X) is closed in X**, where p:X-->X** is defined by p(x)=T_x and T_x:X*-->F is defined by T_x(x*)=x*(x) (F is a field).

I think I should pick a convergent sequence {x_n} in p(X) (x_n -->x)and show that x belongs to p(X). i.e. show there exists a w in X such that p(w)=x.
But for some reason I am not getting the answer.

 

[yyat]

You can use the http://en.wikipedia.org/wiki/Hahn-Banach_theorem" to show that p is an isometry. This then easily implies that its image is closed.