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 $S_1 = TS$.
Pretty please, can someone enlight me ?