I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from

1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton) 2) http://www.math.tamu.edu/~schlump/sofar.pdf (some notes from Professor Schlump)

The proof in 2) is basically the same as in 1) , but with further details. I don't quite get why ker $S$ is contained in Ker $S_1$ . And I don't get why that implies that there is a $T$ such that $S_1 = TS$.

Can someone enlight me ? I would really appreciate it!