Complete measurement space

Complete measurement space ()

Complete SpaceOrComplete measurement spaceIt is a space with the following properties: any sequence of keys in the space converges within the space.

Example

• The rational number space is not complete, because the limited decimal number indicates that it is a Gaussian Sequence, but its limit is not in the rational number space.
• Real Number space is complete
• The open interval (0, 1) is not complete. The sequence (1/2, 1/3, 1/4, 1/5,...) is a kernel sequence, but it does not converge to any (0,
  1.
• LingSIs any set,SNIsSAll sequences in, defineSNUpper sequence (XN) And (YN) Is1/nIf the smallest index existsNFor this index. OtherwiseN0. The measurement space defined in this method is complete. The space is in discrete space.SThe product of several copies.

[Edit] intuitive understanding

Intuitively, a spaceCompleteIt refers to "no holes" and "no skin shortage", both of which are "no disadvantages ". No hole refers to no internal defect, no skin shortage refers to no defect on the boundary. From this point of view, the closure of the same set in a complete space is similar. This is also reflected in the following theorem: The closed subset of a complete space is complete.

[Edit] related Theorem

• Any tightening measurement space is complete. In fact, a measurement space is solid only when the space is complete and completely bounded.
• Any subset of a complete space is complete if and only if it is a closed subset.
• IfXIs a collection,MIs a complete measurement space, then allXMapMBounded FunctionsFB (X,MIs a complete measurement space, where Set B (X,M) Is defined:

• IfXOne topology,MIs a complete measurement space, then allXMapMContinuous Bounded FunctionsFCB (X,M) Is B (X,M) (Defined by the previous objective). Therefore, it is complete.

• Bell's theorem: any complete measurement space is a bell space. That is to say, the Union of the number of nodes with no dense subsets has no interior point.

[Edit] compile [edit] Definition

For any metric spaceMYou can construct a complete measurement space.M'(Or expressed as), making the original measurement space a new dense sub-space of a complete measurement space.M'Has the following universal properties:NFor any complete measurement space,FFor any slaveMToNThe consistent continuous function of has a uniqueM'ToNConsistent Continuous FunctionsF'Make this functionF. Newly constructed complete measurement spaceM'In the sense of equi-width, it is determined by the uniqueness of this property.MOfRegionalization Space.

The above definition is based onMYesM'Concept of dense sub-spaces. We can alsoRegionalization SpaceDefined as includeM. It can be proved that the space defined in this way exists and is unique (in the sense of same distance), and is equivalent to the above definition.

For the switching ring and the model on it, it can also be defined as an ideal completeness and regionalization. For details, refer to entry regionalization.
(Ring theory ).

[Edit] Structure

Similar to the method of defining the irrational number from the rational number field, we can add elements to the original space to make it complete through the kernel sequence.

PairMAny two keys inX =(XN) AndY =(YN), We can define the distance between them:
D (X,Y) = LimND (XN,YN) (This limit exists because the real number field is complete ). The measurement defined in this method is only a pseudo measurement, because different keys can converge to 0. But we can do the same as in many cases (for example, fromLpTo), define the new measurement space as a set of equivalence classes on the set of all the cokeys, where the equivalence class is based on the relationship of 0 distance (it is easy to verify that the relationship is an equivalent relationship ). In this way* X=
{YYesMOn the kernel sequence :},M'= {* X: x
ε M}, Original spaceMTakeX* XThe ing method of is embedded into the new complete measurement space.M'. Easy to verify,MSameM'.

The Consortium consortium Consortium.

[Edit] nature

The real number construction of Conway's is a special case of the above construction. In this case, the real number set can be expressed as the normalization of the absolute value of the rational number set. If another absolute value is obtained in the rational number set, the complete space is the pjin number.

If we apply the preceding process to the norm vector space, we can obtain a space called H, where the original space is a dense space. If it is applied to an inner product space, the obtained result is the Hilbert space, and the original space is still its dense space.

[Edit] related concepts

• Completeness and closure: As mentioned above, completeness is similar to closure. So what is the difference between completeness and closure? The difference between them is that completeness is the nature of space or set, while closed is the nature of subset. Generally, when a set is closed or open, it actually means that the set isR1Or a closed or open subset of a topology. For example, open interval (0,
  1) is the complete set (0, 1) or closed subset, because (0,
  1) The pilot set in these two complete sets is its own. But (0, 1) isR1. The closed subset can be defined by the convergence sequence, because the attention points of the Convergence sequence are always in the complete set. Whether the attention point is in the subset determines whether the subset is a closed subset. In contrast, the definition of completeness does not have the concept of complete sets. This is also why it is necessary to use a kernel sequence instead of a converged sequence in its definition, because there must be a vertices in the definition of a converged sequence, if the vertex is not in the measurement space, the distance between the vertex in the convergence sequence and the vertex is undefined.