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.
Read more about this topic: Discontinuous Linear Map
Famous quotes containing the word concrete:
“A sociosphere of contact, control, persuasion and dissuasion, of exhibitions of inhibitions in massive or homeopathic doses...: this is obscenity. All structures turned inside out and exhibited, all operations rendered visible. In America this goes all the way from the bewildering network of aerial telephone and electric wires ... to the concrete multiplication of all the bodily functions in the home, the litany of ingredients on the tiniest can of food, the exhibition of income or IQ.”
—Jean Baudrillard (b. 1929)
“Babies are necessary to grown-ups. A new baby is like the beginning of all things—wonder, hope, a dream of possibilities. In a world that is cutting down its trees to build highways, losing its earth to concrete ... babies are almost the only remaining link with nature, with the natural world of living things from which we spring.”
—Eda Le Shan (b. 1922)