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 ==