Dynamic Optimization - Infinite dimensional spaces - Reference request Respected community members,
I am currently reading the book "recursive macroeconomic theory" by Sargent and Ljungqvist. While reading this book I have realized that I do not always fully understand what is going on behind "the scenes". 
In particular, in chapter 8, the authors uses the Lagrange method. This method is pretty clear to me in the finite dimensional case, i.e. optimization over $R^n$. However here we are dealing with infinite dimensional problems. Why does the Lagrange method work here? Can someone point me to any good references? 
To clarify: I do understand how to apply the "recipe" given in the book, however I do not understand why it works. What are the specific assumptions needed in order to assure that a solution exists? That it is unique? This is the kind of questions that I would like to be able to answer rigorously. 
I hope I am clear enough about this, if not please let me know. Furthermore I really appreciate any help from you. Thank you for your time. 
Btw: If anyone have the time to look into the matters, the book is available for free here: elogica.br.inter.net/bere/boo/sargent.pdf

A fairly rigorous treatment with many economics applications is Stokey, Lucas and Prescott's (SLP) Recursive Methods in Economic Dynamics. 
This MIT OCW course gives good additional readings. I find the ones on transversality conditions very important.
Standard mathematical treatments are Bertsekas's Dynamic Programming and Optimal Control and Puterman's Markov Decision Processes.
