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As noted above, the [[axiom of choice]] (AC) is used in the general existence theorem of discontinuous linear maps. In fact, there are no constructive examples of discontinuous linear maps with complete ___domain (for example, [[Banach space]]s). In analysis as it is usually practiced by working mathematicians, the axiom of choice is always employed (it is an axiom of [[ZFC]] [[set theory]]); thus, to the analyst, all infinite dimensional topological vector spaces admit discontinuous linear maps.
On the other hand, in 1970 [[Robert M. Solovay]] exhibited a [[model (model theory)|model]] of [[set theory]] in which every set of reals is measurable.<ref>{{
| last = Solovay | first = Robert M. | authorlink = Robert M. Solovay
| journal = [[Annals of Mathematics]]
| mr = 0265151
| pages = 1–56
| series = Second Series
| title = A model of set-theory in which every set of reals is Lebesgue measurable
| volume = 92
| year = 1970}}.</ref> This implies that there are no discontinuous linear real functions. Clearly AC does not hold in the model.
Solovay's result shows that it is not necessary to assume that all infinite-dimensional vector spaces admit discontinuous linear maps, and there are schools of analysis which adopt a more [[constructivism (mathematics)|constructivist]] viewpoint. For example H. G. Garnir, in searching for so-called "dream spaces" (topological vector spaces on which every linear map into a normed space is continuous), was led to adopt ZF + [[dependent choice|DC]] + [[Baire property|BP]] (dependent choice is a weakened form and the [[Baire property]] is a negation of strong AC) as his axioms to prove the [[Garnir–Wright closed graph theorem]] which states, among other things, that any linear map from an [[F-space]] to a TVS is continuous. Going to the extreme of [[Constructivism (mathematics)|constructivism]], there is [[Ceitin's theorem]], which states that ''every''
The upshot is that it is not possible to obviate the need for AC; it is consistent with set theory without AC that there are no discontinuous linear maps. A corollary is that constructible discontinuous operators such as the derivative cannot be everywhere-defined on a complete space.
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