Bounded Operator - Linearity and Boundedness

Linearity and Boundedness

Not every linear operator between normed spaces is bounded. Let X be the space of all trigonometric polynomials defined on, with the norm

Define the operator L:X→X which acts by taking the derivative, so it maps a polynomial P to its derivative P′. Then, for

with n=1, 2, ...., we have while as n→∞, so this operator is not bounded.

It turns out that this is not a singular example, but rather part of a general rule. Any linear operator defined on a finite-dimensional normed space is bounded. However, given any normed spaces X and Y with X infinite-dimensional and Y not being the zero space, one can find a linear operator which is not continuous from X to Y.

That such a basic operator as the derivative (and others) is not bounded makes it harder to study. If, however, one defines carefully the domain and range of the derivative operator, one may show that it is a closed operator. Closed operators are more general than bounded operators but still "well-behaved" in many ways.