Complete metric space

A complete space is a metric space analysis in which converges every Cauchy sequence of elements of the space. For example, the space of rational numbers with the sum metric is not complete, because as the number is not rational, but it Cauchy sequences of rational numbers is that converge at embedding of the rational numbers in the real numbers against and thus does not violate any rational number. However, it is always possible to fill the holes, thus completing an incomplete metric space. In the case of rational numbers obtained by the space of real numbers.

Definition

A sequence of elements of a metric space is called Cauchy sequence if

Applies. Next a sequence converges to an element, if

Applies.

A metric space is now called complete if every Cauchy sequence converges in him.

Comments

• It is true that a convergent sequence is always a Cauchy sequence, but the reverse direction does not need to be necessarily true. In a full room now has a sequence exactly then a threshold, if it is a Cauchy sequence; So the two terms coincide.

• Often there are calls in the definition of completeness that every Cauchy sequence converges to an element "in". The addition of "in" is not necessary, as for consequences already in accordance with the definition of convergence only elements as Limits come into question.

Examples

• The set of rational numbers is the amount of metric

• The closed real interval, the set of real numbers and the set of complex numbers are each completely with the real or complex amount metric.

• The open interval is the amount of real metric is not complete, because the limit value of the harmonic sequence does not lie in the interval. However, there is complete metrics that produce the same topology as the amount of metrics, for example,

• The space of p- adic numbers for each prime is complete. This space is the completion of with respect to the metric of the p- adic amount

• Every finite scalar product, for example the Euclidean vector space or the unitary vector space with the standard scalar product, with the derived from the scalar metric

• Every finite- normed space, for example, the space of real or complex matrices or a matrix norm, with the derived from the standard metric

• If any non-empty set, then you can make the set of all sequences in a complete metric space by the distance between two sequences is to

• For more examples of complete spaces of infinite dimension, see the articles Banach space and Hilbert space.

Some sets

Every compact metric space is complete. A metric space is compact if it is complete and totally bounded.

A subset of a complete space is even if and only complete if it is complete.

If a non-empty set and a complete metric space, then the space of bounded functions from to with the metric

A complete metric space.

If a topological space and a complete metric space, then the set of bounded continuous functions from to a closed subset of and as such with the above metric is complete.

Completion

Every metric space with a metric can be completed, that is, there is a complete metric space with a metric and an isometry, so that is dense in. The space is called completion. Since all completions are isomorphic by isometric, one also speaks of the completion.

Construction

The completion of one can construct a set of equivalence classes of Cauchy sequences in. We define the distance between two Cauchy sequences and by

This distance is well defined, but it is only a pseudo- metric, for different Cauchy sequences can have the distance. The property is an equivalence relation on the set of Cauchy sequences and the set of all equivalence classes is with this notion of distance is a complete metric space. If we identify each element of the equivalence class of the constant sequence in, we obtain an isometric embedding in.

If a normed space, we can also make it easier its completion by

Selected as the completion of the image of the Bidualraum under the canonical embedding.

Properties

Cantor's construction of the real numbers from the rational is a special case thereof. As I said above, one obtains other metric spaces, if one uses instead of the usual amount metric p -adic metric and completed.

Completes you a normed vector space, we obtain a Banach space that contains the original space as a dense subspace. Therefore, you also get a Hilbert space, if one completes a Euclidean vector space, because the parallelogram law is fulfilled as a normed space and the complete inner product is then obtained via the polarization formula in the completion.

Uniformly continuous mappings of a metric space into a complete metric space can always be uniquely ( automatically also uniformly ) continuous maps continue to completion with values ​​in.

Completely metrizable spaces

Completeness is a property of the metric, not the topology, that is, a complete metric space may be homeomorphic to an incomplete metric space. For example, the real numbers are complete, but homeomorphic to an open interval is not complete ( for example, a homeomorphism from to). Another example is the irrational numbers, while not complete, but is homeomorphic to the area of ​​natural number sequences ( an example of a special case above).

In the topology is considered fully metrizable spaces, i.e. spaces for which there is at least one complete metric that generates the existing topology.

Uniform spaces

Like many other concepts from the theory of metric spaces can also generalize the notion of completeness to the class of uniform spaces: A uniform space is called complete if every Cauchy net converges. The most of the above statements remain valid in the context of uniform spaces, for example, also has every uniform space a unique completion.

Topological vector spaces carry a natural uniform structure and they are called complete if it is complete with respect to this uniform structure. They are called quasi-complete if every bounded Cauchy net converges, that is, if every bounded closed set is complete.