Dimension (mathematics)

In that Mathematics becomes with that Dimension a concept marks, that essentially the amount of that Degrees of freedom one Movement in a certain Area designated.

Definitions

The term of the dimension arises in a multiplicity of connections. No individual mathematical concept is able it, to define the dimension for all situations satisfyingly, therefore also different dimension terms exist for different areas.

Hamel dimension

The dimension one is most well-known Vector space, also Hamel dimension called. It is equal the number that Basis vectors the vector space, thus equal the power of a minimum Production system.

For example the geometrically descriptive possesses Euclidean 3-Raum the dimension 3 (length, Width, Height). That corresponds to the area, in which we move ourselves us and is the highest dimension, which we still descriptive to introduce itself can. The Euclidean level has the dimension 2; those Number line the dimension 1, that Point the dimension 0.

Vector spaces, no finite production system possess, one can do likewise those Power a minimum production system as dimension assign; it concerns thereby then an infinite Cardinal number.

The dimensiondimension the dimension

Accordingly one knows the power of one The basisbasis the basis a topological vector space (in particular one Hilbertraums) designate also as dimension.

Variousnesses

Besides the dimension is one Diversity likewise descriptive obvious. By definition each point of a diversity has one Environment, those homoeomorph to the n-dimensional Euclidean area is; this n is called dimension of the diversity. In order to prevent, that the dimension depends on the choice of the point, becomes the dimension term usually only for coherent Variousnesses uses.

Well-known two-dimensional variousnesses is the surface of a ball or that Moebiusband.

Chain length as dimension (topological dimension)

The dimension of a vector space is equal to the maximum length (number of inklusionen) of a chain of into one another contained Unterraeumen. The aspect of the dimension as chain length permits a verallgemeinerung on other structures.

So is about those Krulldimension a commutative Ring as maximum length of a chain of into one another contained Prime ideals defined.

Likewise the dimension of a diversity is the maximum length of a chain of into one another contained variousnesses, with the each member the chain Edge a subset of the previous is. For example the edge of the globe is the earth's surface; Edge of their subset Germany is the state border; Edge of a certain border section are the two terminator points - there it no longer chain gives, the globe has dimension 3. Since inklusion and edge formation are always defined, this supplies a dimension term for everyone topological area (so-called inductive dimension).

Hausdorff dimension

Apart from the integral dimensions indicated so far one knows also generalized, rationally- or real zahlige Dimensions, with their assistance the in such a way specified Fraktale are quantified. An example is those Hausdorff dimension.

See also: Dimension (physics)