discontinuous linear map

Discontinuous linear maps can be proven to exist more generally even if the space is complete.

If the given operator is not bounded then the extension is a discontinuous linear map.

On every infinite-dimensional topological vector space there is a discontinuous linear map.

This is however false for any infinite-dimensional normed space, as shown by the example of discontinuous linear maps.

In fact, there are no constructive examples of discontinuous linear maps with complete domain (for example, Banach spaces).

As noted above, the axiom of choice (AC) is used in the general existence theorem of discontinuous linear maps.

Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at Discontinuous linear map and based on the axiom of choice.

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).

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.

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.

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.

As a consequence of the Stone-Weierstrass theorem, the graph of this operator is dense in XxY, so this provides a sort of maximally discontinuous linear map (confer nowhere continuous function).

Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence of independent vectors which does not have a limit, a linear operator may grow without bound.

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.

The argument for the existence of discontinuous linear maps on normed spaces can be generalized to all metrisable topological vector spaces, especially to all Fréchet-spaces, but there exist infinite dimensional locally convex topological vector spaces such that every functional is continuous.