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*](https://books.google.ch/books/about/Analyse_fonctionnelle.html?id=5VuVxgEACAAJ&redir_esc=y).

Also in most of my monographs, in particular [Topological Analysis](https://www.amazon.de/Topological-Analysis-Nonlinear-Inclusions-Applications-ebook/dp/B07G4PFDHS), 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)](https://www.sciencedirect.com/science/article/abs/pii/S0168007221000178)