Discontinuous linear map: Difference between revisions

Revision as of 21:21, 27 January 2014

Revision as of 01:41, 9 March 2014

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In [[mathematics]], [[linear map]]s form an important class of "simple" [[function (mathematics)|functions]] which preserve the algebraic structure of [[linear space]]s and are often used as approximations to more general functions (see [[linear approximation]]). If the spaces involved are also [[topological space]]s (that is, [[topological vector space]]s), then it makes sense to ask whether all linear maps are [[continuous map|continuous]]. It turns out that for maps defined on infinite-[[dimension (linear algebra)|dimensional]] topological vector spaces (e.g., infinite-dimensional [[normed space]]s), the answer is generally no: there exist '''discontinuous linear maps'''. If the ___domain of definition is [[complete space|complete]], such maps can be proven to exist, but the proof relies on the [[axiom of choice]] and does not provide an explicit example.

== A linear map from a finite -dimensional space is always continuous ==

Let ''X'' and ''Y'' be two normed spaces and ''f'' a linear map from ''X'' to ''Y''. If ''X'' is [[finite-dimensional]], choose a base (''e''₁, ''e''₂, …, ''e''_''n'') in ''X'' which may be taken to be unit vectors. Then,

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Thus, ''f'' is a [[bounded linear operator]] and so is continuous.

If ''X'' is infinite-dimensional, this proof will fail as there is no guarantee that the [[supremum]] ''M'' exists. If ''Y'' is the zero space {0}, the only map between ''X'' and ''Y'' is the zero map which is trivially continuous. In all other cases, when ''X'' is infinite -dimensional and ''Y'' is not the zero space, one can find a discontinuous map from ''X'' to ''Y''.

== A concrete example ==

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== Axiom of choice ==

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>{{citation

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== Beyond normed spaces ==

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. On the other hand, the [[Hahn–Banach theorem]], which applies to all locally convex spaces, guarantees the existence of many continuous linear functionals, and so a large dual space. In fact, to every convex set, the [[Minkowski gauge]] associates a continuous [[linear functional]]. The upshot is that spaces with fewer convex sets have fewer functionals, and in the worst case scenario, a space may have no functionals at all other than the zero functional. This is the case for the [[Lp space|''L''^''p''('''R''',''dx'')]] spaces with 0&nbsp;<&nbsp;''p''&nbsp;<&nbsp;1, from which it follows that these spaces are nonconvex. Note that here is indicated the [[Lebesgue measure]] on the real line. There are other ''L''^''p'' spaces with 0&nbsp;<&nbsp;''p''&nbsp;<&nbsp;1 which do have nontrivial dual spaces.

Another such example is the space of real-valued [[measurable function]]s on the unit interval with [[quasinorm]] given by