Another library branch that I think is even closer than Playa Vista is the
Westchester – Loyola Village Branch Library
7114 W. Manchester Avenue
Los Angeles, 90045
Here are the hours is is open:
Mondays 10:00 am – 8:00 pm
Tuesdays Noon – 8:00 pm
Wednesdays 10:00 am – 8:00 pm
Thursdays Noon – 8:00 pm
Fridays 10:00 am – 6:00 pm
Saturdays 10:00 am – 6:00 pm
Sunday CLOSED
It is on the corner of Lincoln and Manchester and Lincoln runs right into LAX at the point. It also has the advantage of being kitty corner across the street from the OTIS school of design where you can usually find a safe place to study after the librairies close. And it has a nice little inexpensive crepe/coffee place that is just a bit further down Lincoln (and lots of other restaurants nearby as well) :
Kc’s Cafe
8320 Lincoln Blvd
Los Angeles, CA 90045–2466
Phone: (310) 348–9697
Here is a link to a map of the library-which is almost right on the runway :)
http://www.lapl.org/maps/branch_googlemap.php?branchid=13