Editing Unbounded operator (section)

{{Short description|Linear operator defined on a dense linear subspace}}
In [[mathematics]], more specifically [[functional analysis]] and [[operator theory]], the notion of '''unbounded operator''' provides an abstract framework for dealing with [[differential operator]]s, unbounded [[observable]]s in [[quantum mechanics]], and other cases.

The term "unbounded operator" can be misleading, since
* "unbounded" should sometimes be understood as "not necessarily bounded";
* "operator" should be understood as "[[linear operator]]" (as in the case of "bounded operator");
* the domain of the operator is a [[linear subspace]], not necessarily the whole space;
* this linear subspace is not necessarily [[closed set|closed]]; often (but not always) it is assumed to be [[dense (topology)|dense]];
* in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.

In contrast to [[bounded operator]]s, unbounded operators on a given space do not form an [[algebra over a field|algebra]], nor even a linear space, because each one is defined on its own domain.

The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above.