Q:

Rigorous proof of quantum electrodynamics renormalization

In most physics books they give  proofs of renormalization of quantum electrodynamics that are not  mathematically rigorous. 
Is there any book or article that give a formal proof of quantum electrodynamics renormalization in the mathematical sense? 

A:

The comments above conflate two different kinds of constructions or proofs of renormalizability: perturbative and nonperturbative.  Perturbative means that correlation functions are constructed as formal power series in some meaningless symbol which pretends to be a parameter like Planck's constant or the fine structure constant.  Nonperturbative means a full construction including a resummation of this power series to give a true function of this parameter with values in say Schwartz distributions.  For QED in 4d there is no rigorous nonperturbative construction.  In fact it is believed to be impossible (triviality conjecture).  However, rigorous perturbative renormalization proofs exist.  The most mathematically careful reference I know is: "QED: A proof of Renormalizability" by Feldman, Hurd and Rosen.