Discontinuous linear map: Difference between revisions

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| doi=10.2307/1970696}}.</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'' function is continuous (this is to be understood in the terminology of constructivism, according to which only representable functions are considered to be functions).<ref>{{citation|title=Handbook of Analysis and Its Foundations|first=Eric|last=Schechter|publisher=Academic Press|year=1996|isbn=9780080532998|page=136|url=httphttps://books.google.com/books?id=eqUv3Bcd56EC&pg=PA136}}.</ref> Such stances are held by only a small minority of working mathematicians.

The upshot is that the existence of discontinuous linear maps depends on AC; it is consistent with set theory without AC that there are no discontinuous linear maps on complete spaces. In particular, no concrete construction such as the derivative can succeed in defining a discontinuous linear map everywhere on a complete space.