Discontinuous Linear Map - Concrete Example

Concrete Example

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 a sense, the linear operators are not continuous because the space has "holes".

For example, consider the space X of real-valued smooth functions on the interval with the uniform norm, that is,

The derivative at a point map, given by

defined on X and with real values, is linear, but not continuous. Indeed, consider the sequence

for n≥1. This sequence converges uniformly to the constantly zero function, but

as n→∞ instead of which would hold for a continuous map. Note that T is real-valued, and so is actually a linear functional on X (an element of the algebraic dual space X*). The linear map X → X which assigns to each function its derivative is similarly discontinuous. Note that although the derivative operator is not continuous, it is closed.

The fact that the domain is not complete here is important. Discontinuous operators on complete spaces require a little more work.