# Closed in a dual space

• Mar 1st 2009, 10:17 AM
math8
Closed in a dual space
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.
• Mar 1st 2009, 01:21 PM
Opalg
Quote:

Originally Posted by 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.

For x in X, $\|p(x)\| = \sup\{\|T_x(x^*)\|:x^*\in X^*,\;\|x^*\|=1\} = \sup\{\|x^*(x)\|:\|x^*\|=1\} = \|x\|$ by the Hahn–Banach theorem. So the map p is isometric, and it follows easily from that that its range must be closed.