Home > Completeness

This is made precise in various ways, several of which have a related notion of completion.

• Metric spaces or uniform spaces are said to be complete if every Cauchy sequence in them converges. See complete space.

• In functional analysis, a subset S of a topological vector space V is complete if its span is dense in V. If V is separable, it follows that any vector in V can be written as a (possibly infinite) linear combination of vectors from S. In the particular case of Hilbert spaces (or more generally, inner product spaces), an orthonormal basisIn mathematics, an orthonormal basis of an inner product space V i. a vector space with an inner product), or in particular of a Hilbert space H is a set of elements whose span is dense in the space, in which the elements are mutually orthogonal and norma is a set that is both complete and orthonormal.

• A measure spaceMeasure theory In mathematics, a measure is a function that assigns a number, e. a "size", "volume", or "probability", to subsets of a given set. The concept is important in mathematical analysis and probability theory. Measure theory is that branch of re is complete if every subset of every null setIn measure theory, a null set is a set that it is negligible for the purposes of the measure in question. Which sets are null will depend on the measure considered. Thus one may speak of m-null sets for a given measure m. The term "null set" is sometimes is measurable. See complete measureIn measure theory, a complete measure is a measure in which every subset of every null set is measurable (having measure 0). Every measure has an extension that is complete. The smallest such extension is called the completion of the measure. Suppose μ.

• In statisticsStatistics is the science and practice of developing human knowledge through the use of empirical data. It is based on statistical theory which is a branch of applied mathematics. Within statistical theory, randomness and uncertainty are modelled by proba, a statisticA statistic (singular) is the result of applying a statistical algorithm to a set of data. In the calculation of the arithmetic mean, for example, the algorithm directs us to sum all the data values and divide by the number of data items. In this case, we is called complete if it does not allow an unbiased estimator of zero. See completeness (statistics)Suppose a random variable X (which may be a sequence X . X of scalar-valued random variables), has a probability distribution belonging to a known family of probability distributions, parametrized by θ, which may be either vector- or scalar-valued..

• In graph theory, a complete graph is an undirected graph where every pair of vertices has exactly one edge connecting them.

• In category theory, a category C is called complete if every functor from a small category to C has a limit; it is called cocomplete if every such functor has a colimit. For more information, see the given article on limits in category theory.

• In order theory and related fileds such as lattice and domain theory, completeness generally refers to the existence of certain suprema or infima of some partially ordered set. Notable special usages of the term include the concepts of complete lattice and complete partial order (cpo). Furthermore, an ordered field is complete if every non-empty subset of it that has an upper bound within the field has a least upper bound within the field, which should be compared to the (slightly different) order theoretical notion of bounded completeness. Up to isomorphism there is only one complete ordered field: the field of real numbers.

• In logic, a formal calculus (often just specified by a set of additional axioms used to formalize some theory within the underlying logic) is said to be complete if, for any statement P, a proof exists for P or for not P. A system is consistent if a proof never exists for both P and not P. Gödel's incompleteness theorem proved that no system as powerful as the Peano axioms can be both consistent and complete. See also below for another notion of completeness in logic.

• In proof theory and related fields of mathematical logic, a formal calculus is said to be complete with respect to a certain logic (i.e. with respect to its semantics), if every statement P that follows semantically from a set of premises G can be derived syntactically from these premisses within the calculus. Formally, implies . Especially, all tautologies of the logic can be proven. Even when working with classical logic, this is not equivalent to the notion of completeness introduced above (both a statement and its negation might not be tautologies with respect to the logic). The reverse implication is called soundness.

• In computational complexity theory, a problem P is said to be complete for a complexity class C, under a given type of reduction, if P is in C, and every problem in C reduces to P using that reduction. For example, each problem in the class NP-Complete is complete for the class NP, under polynomial-time, many-one reduction.