One can define an extended norm on the space where is a subset of as follows:
for all because the -th derivative of is again , which blows up as approaches infinity. If we are considering functions on a compact (closed and bounded) subset of however, the norm is always finite as a consequence of the fact that every continuous function on a compact set attains a maximum. This also means that we may replace the “” with a “” in our definition in this case.
Having a sequence of functions converge under this norm is the same as having their -th derivatives converge uniformly. Therefore, it follows from the fact that the uniform limit of continuous functions is continuous that is complete under this norm. (In other words, it is a Banach space.)
In the case of , there is no natural way to impose a norm, so instead one uses all the norms to define the topology in . One does this by declaring that a subset of is closed if it is closed in all the norms. A space like this whose topology is defined by an infinite collection of norms is known as a multi-normed space.
|Date of creation||2013-03-22 14:59:46|
|Last modified on||2013-03-22 14:59:46|
|Last modified by||rspuzio (6075)|