Everyone should begin with something in the line of the following five
texts: Torchinsky’s Real Variables; Dummit and Foote’s Abstract Algebra; Do
Carmo’s Differential Geometry; Hoffman and Kunze’s Linear Algebra; and
Ahlfors’ Complex Analysis. [I included a collection of links to the books
below for your convenience] I know, I know: there are much better sources
than these out there. The ones I presented have an advantage over any other,
in my opinion: a huge selection of problems. If you patiently solve every
single one of them, none of these topics will hold secrets for you. That is
a promise.

Mine includes “Variational Analysis” (Rockafellar, Wets), “Linear Operators
(Part I)” (Dunford, Schwartz – only partly…), “Regularization of ill-posed
problems” (Engl, Hanke, Neuabuer), “Nonsmooth Analysis” (Schirotzek)…

 

http://www.amazon.com/gp/product/0817637206/ref=as_li_tf_tl?ie=UTF8&tag=blancosilva-20&linkCode=as2&camp=1789&creative=9325&creativeASIN=0817637206