On the relation of three theorems of analysis to the axiom of choice

In what follows, essentially two things will be accomplished: Firstly, it will be proven that a version of the Arzel`a–Ascoli theorem and the Fr'echet–Kolmogorov theorem are equivalent to the axiom of countable choice for subsets of real numbers. Secondly, some progress is made towards determining the amount of axioms that have to be added to the Zermelo–Fraenkel system so that the uniform boundedness principle holds.