Dimension

In mathematics, a concept is referred to as the dimension of which essentially indicates the number of degrees of freedom of movement in a certain space.

The concept of dimension occurs in a variety of contexts. No single mathematical concept is able to define the dimension for all situations satisfactorily, therefore exist for different rooms, different dimension terms.

Hamel dimension ( dimension of a vector space )

Best known the dimension of a vector space, also Hamel dimension is called. It is equal to the cardinality of a basis of the vector space. The following statements are equivalent for this purpose:

• The dimension is equal to the cardinality of a minimal generating system.
• The dimension is equal to the cardinality of a maximal system of linearly independent vectors.

For example, the geometrically intuitive Euclidean 3- space has dimension 3 (length, width, height). The Euclidean plane has dimension 2, the number is precisely the dimension 1, the point of the dimension 0

Vector spaces that have no finite generating system, you can also assign the cardinality of a minimal generating system as a dimension; then it involves an infinite cardinal number.

The word " Hamel basis " is used mainly for infinite-dimensional vector spaces, because Georg Hamel has proven to be the first (using the well-ordering theorem, so the axiom of choice ) the existence of a basis in this case.

Hilbert space dimension

Every Hilbert space has an orthonormal basis. Only when these finite number of elements, it is a Hamel basis in the sense defined above. One can show that any two orthonormal bases have the same number of elements, and thus it is possible to define the dimension of the Hilbert space as the cardinality of an orthonormal basis; It is also this to be a cardinal number. This cardinal number is sufficient to classify complete Hilbert spaces: for every cardinal number up to isomorphism exactly one Hilbert space has an orthonormal basis of the corresponding cardinality.

For example, the Hilbert space of square integrable functions on [0, 1] Hilbert space dimension - the Hamel dimension, however, is strictly greater.

Manifolds

In addition, the dimension of a manifold is also clearly understood. By definition, each point of a manifold has a neighborhood which is homeomorphic to the - dimensional Euclidean space; this is called dimension of the manifold. In order to prevent the dimension of the choice of the point depends on the dimension term is usually used only for connected manifolds or manifolds are defined from the outset so that the model space and thus the dimension are the same everywhere.

Well-known two-dimensional manifolds are the surface of a sphere or the Möbius band.

Chain length as a dimension ( topological dimension)

The dimension of a vector space is equal to the maximum length (number of inclusions ) of a chain of sub-spaces contains one another. The perception of the dimension as chain length allows for a generalization to other structures.

Thus, the Krull dimension of a commutative ring is approximately defined as the maximum length of a chain of prime ideals contained in one another minus 1.

Similarly, the dimension of a manifold is the maximum length of a chain of interlocking contained manifolds, each term is the chain edge of a subset of the previous one. For example, the edge of the earth is the earth's surface; Edge of their subset Germany is the state border; Edge of a certain section of the border, the two end points - there is no longer the chain, the globe has dimension 3, since inclusion and fringing are always defined, it provides a notion of dimension for every topological space ( the so-called inductive dimension). Another topological concept of dimension is the Lebesgue covering dimension.

Fractal dimension

In addition to the integer dimensions previously mentioned is also known generalized, rational or reellzahlige dimension terms, with the help of so-called fractals can be compared.