My lectures are largely based on the lecture notes (in French) by Bernard Maurey:
Here are some classical textbooks (only the first chapters are relevant for this class):
Functional analysis, Sobolev spaces and partial differential equations.
Universitext. Springer, New York, 2011. xiv+599 pp
Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991
Reprint of the sixth (1980) edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xii+501 pp