Editing Real analysis (section)

===Order properties of the real numbers===
The real numbers have various [[lattice theory|lattice-theoretic]] properties that are absent in the complex numbers. Also, the real numbers form an [[ordered field]], in which sums and products of positive numbers are also positive. Moreover, the ordering of the real numbers is [[totally ordered|total]], and the real numbers have the [[least upper bound property]]: <blockquote>''Every nonempty subset of <math>\mathbb{R}</math> that has an upper bound has a [[Supremum|least upper bound]] that is also a real number.'' </blockquote>These [[partially ordered set|order-theoretic]] properties lead to a number of fundamental results in real analysis, such as the [[monotone convergence theorem]], the [[intermediate value theorem]] and the [[mean value theorem]].

However, while the results in real analysis are stated for real numbers, many of these results can be generalized to other mathematical objects. In particular, many ideas in [[functional analysis]] and [[operator theory]] generalize properties of the real numbers – such generalizations include the theories of [[Riesz space]]s and [[positive operator]]s. Also, mathematicians consider [[real part|real]] and [[imaginary part]]s of complex sequences, or by [[strong operator topology|pointwise evaluation]] of [[operator (mathematics)|operator]] sequences.{{Clarify|date=June 2021}}