It turns out that for maps defined on infinite-dimensional topological vector spaces (e.g., infinite-dimensional normed spaces), the answer is generally no: there exist discontinuous linear maps.
If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example.
for some C>0 which follows from the fact that any two norms on a finite-dimensional space are equivalent, one finds
If X is infinite-dimensional, this proof will fail as there is no guarantee that the supremum M exists.
Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence
In a sense, the linear operators are not continuous because the space has "holes".
of real-valued smooth functions on the interval [0, 1] with the uniform norm, that is,
f acts as the identity on the rest of the Hamel basis, and extend to all of
This example can be extended into a general theorem about the existence of discontinuous linear maps on any infinite-dimensional normed space (as long as the codomain is not trivial).
Discontinuous linear maps can be proven to exist more generally, even if the space is complete.
If X is infinite-dimensional, to show the existence of a linear functional which is not continuous then amounts to constructing f which is not bounded.
T so defined will extend uniquely to a linear map on X, and since it is clearly not bounded, it is not continuous.
Notice that by using the fact that any set of linearly independent vectors can be completed to a basis, we implicitly used the axiom of choice, which was not needed for the concrete example in the previous section.
As noted above, the axiom of choice (AC) is used in the general existence theorem of discontinuous linear maps.
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.
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 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 + DC + 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.
[2] 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.
It makes sense to ask which linear operators on a given space are closed.
So the natural question to ask about linear operators that are not everywhere-defined is whether they are closable.
The answer is, "not necessarily"; indeed, every infinite-dimensional normed space admits linear operators that are not closable.
As in the case of discontinuous operators considered above, the proof requires the axiom of choice and so is in general nonconstructive, though again, if X is not complete, there are constructible examples.
In fact, there is even an example of a linear operator whose graph has closure all of
so this provides a sort of maximally discontinuous linear map (confer nowhere continuous function).
It illustrates the fact that an extra dose of caution is needed in doing analysis on infinite-dimensional spaces as compared to finite-dimensional ones.
In fact, to every convex set, the Minkowski gauge associates a continuous linear functional.
Another such example is the space of real-valued measurable functions on the unit interval with quasinorm given by
For example, the existence of a homomorphism between complete separable metric groups can also be shown nonconstructively.