Discontinuous linear map: Difference between revisions

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Examples of discontinuous linear maps are easy to construct in spaces that are not complete; on any Cauchy sequence <math>e_i</math> of linearly independent vectors which does not have a limit, there is a linear operator <math>T</math> such that the quantities <math>\|T(e_i)\|/\|e_i\|</math> grow without bound. In a sense, the linear operators are not continuous because the space has "holes".

For example, consider the space ''<math>X''</math> of real-valued [[smooth function]]s on the interval [0, 1] with the [[uniform norm]], that is,

<math display=block>\|f\| = \sup_{x\in [0, 1]}|f(x)|.</math>

The ''[[derivative]]-at-a-point'' map, given by

<math display=block>T(f) = f'(0)\,</math>

defined on ''<math>X''</math> and with real values, is linear, but not continuous. Indeed, consider the sequence

<math display=block>f_n(x)=\frac{\sin (n^2 x)}{n}</math>

for <math>n \geq 1.</math>. This sequence converges uniformly to the constantly zero function, but

<math display=block>T(f_n) = \frac{n^2\cos(n^2 \cdot 0)}{n} = n\to \infty</math>

as <math>n \to \infty</math> instead of <math>T(f_n)\to T(0)=0</math>, whichas would hold for a continuous map. Note that ''<math>T''</math> is real-valued, and so is actually a [[linear functional]] on ''<math>X''</math> (an element of the algebraic [[dual space]] ''X''<supmath>X^*</supmath>). The linear map ''<math>X'' →\to ''X''</math> which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is [[closed operator|closed]].

The fact that the ___domain is not complete here is important.: Discontinuousdiscontinuous operators on complete spaces require a little more work.

== A nonconstructive example ==

An algebraic basis for the [[real number]]s as a vector space over the [[rationals]] is known as a [[Hamel basis]] (note that some authors use this term in a broader sense to mean an algebraic basis of ''any'' vector space). Note that any two [[commensurability (mathematics)|noncommensurable]] numbers, say 1 and <math>\pi</math>, are linearly independent. One may find a Hamel basis containing them, and define a map <math>f : \R \to \R</math> so that <math>f(\pi) = 0,</math> ''f'' acts as the identity on the rest of the Hamel basis, and extend to all of <math>\R</math> by linearity. Let {''r''_''n''}_''n'' be any sequence of rationals which converges to <math>\pi</math>. Then lim_''n'' ''f''(''r''_''n'') = π, but <math>f(\pi) = 0.</math> By construction, ''f'' is linear over <math>\Q</math> (not over <math>\R</math>), but not continuous. Note that ''f'' is also not [[measurable function|measurable]]; an [[Additive map|additive]] real function is linear if and only if it is measurable, so for every such function there is a [[Vitali set]]. The construction of ''f'' relies 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).

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== General existence theorem ==

Discontinuous linear maps can be proven to exist more generally, even if the space is complete.{{clarify|reason=A general "constructive" proof in the incomplete case was not given above, so this contrast seems kind of hand-wavey.|date=May 2015}} Let ''X'' and ''Y'' be [[normed space]]s over the field ''K'' where <math>K = \R</math> or <math>K = \Complex.</math> Assume that ''X'' is infinite-dimensional and ''Y'' is not the zero space. We will find a discontinuous linear map ''f'' from ''X'' to ''K'', which will imply the existence of a discontinuous linear map ''g'' from ''X'' to ''Y'' given by the formula <math>g(x) = f(x) y_0</math> where <math>y_0</math> is an arbitrary nonzero vector in ''Y''.

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. For that, consider a [[sequence]] (''e''_''n'')_''n'' (<math>n \geq 1</math>) of [[linearly independent]] vectors in ''X'', which we normalize. DefineThen, we define

<math display=block>T(e_n) = n\|e_n\|\,</math>

for each <math>n = 1, 2, \ldots</math> Complete this sequence of linearly independent vectors to a [[basis (vector space)|vector space basis]] of ''X'', andby definedefining ''T'' at the other vectors in the basis to be zero. ''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 but one.

== Role of the axiom of choice ==

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| volume = 92

| year = 1970

| issue = 1 | doi=10.2307/1970696| jstor = 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=https://books.google.com/books?id=eqUv3Bcd56EC&pg=PA136}}.</ref> Such stances are held by only a small minority of working mathematicians.

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* {{annotated link|Finest locally convex topology}}

* {{annotated link|Sublinear function}}

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==Notes==

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== References ==

{{reflist}}

* Constantin Costara, Dumitru Popa, ''Exercises in Functional Analysis'', Springer, 2003. {{isbn|1-4020-1560-7}}.

* [[Eric Schechter|Schechter, Eric]], ''Handbook of Analysis and its Foundations'', Academic Press, 1997. {{isbn|0-12-622760-8}}.

{{Functional Analysisanalysis}}

{{Topological vector spaces}}

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[[Category:Functional analysis]]

[[Category:Axiom of choice]]

[[Category:Functions and mappings]]