One of the standard texts which presents functional analysis only based on ZF+DC is the monograph (consisting of 3 volumes) Henry G. Garnir, Marc de Wilde, and Jean Schmets, Analyse Fonctionnelle.
Also in most of my monographs, in particular Topological Analysis, you will find many of the standard results of analysis and topology with explicit notes for which parts of the assertions more than ZF+DC is needed (and in a few cases also remarks when ZF alone is sufficient). Also in those of my monographs more related with integration and measure theory no use of anything more than ZF+DC is made unless explicitly mentioned. For nonstandard analysis the situation is different, although there are some recent papers that a certain internal nonstandard analysis can be carried out in ZF(+DC) as well.¹
In pure ZF (without DC) most of analysis is known to break down, in particular, it is almost impossible to do a reasonable measure or integration theory (as the real line might be a countable union of countable sets) or even topology (since sequential and topological definitions of a limit can differ already for functions of the real line).
Edit: ¹See e.g. Karel Hrbacek, Mikhail G. Katz, Infinitesimal analysis without the Axiom of Choice, Annals Pure Appl. Logic 172 (2021) (6)