Hyperplane

• 2.1 Functional Analysis ( in beliebigdimensionalen real spaces )
• 2.2 Geometry ( in finite dimensional spaces )

• 3.1 Classical Geometry
• 3.2 Finite Geometry

Definitions

Linear Algebra and Functional Analysis

Let V be a vector space of arbitrary dimension over a field K. Then is called a proper subspace linear hyperplane in V if one of the following equivalent conditions is satisfied:

These statements remain even then equivalent if K is a division ring and V is a left vector space over K. This generalization is important for the subsequent definition of the analytic geometry.

Analytic geometry

If A is a desarguesscher, n- dimensional affine space over a division ring K and the corresponding K- Links vector space of translations ( parallel shifts ) of A, then iff is a hyperplane in A if a ( linear -dimensional ) hyperplane exists, so that for a fixed point is valid.

If P is a desarguesscher, n- dimensional projective space over a division ring K and the associated koordinatisierende Links vector space, then iff is a hyperplane in A if a (linear, n- dimensional ) hyperplane exists, the koordinatisiert exactly H.

See for the generalization of the affine and projective planes nichtdesarguessche concept to the section # geometry ( in finite dimensional spaces ) in this article.

Properties

Functional Analysis ( in beliebigdimensionalen real spaces )

An important relationship exists between hyperplanes topological vector spaces and linear functionals.

• The core of each linear functional that is not 0 is a hyperplane.
• For each hyperplane exists a linear functional, so that the core of the functional is the hyperplane.
• A functional is continuous if the core is complete.
• A hyperplane is either completed or is close in the vector space.

• In the one-dimensional space are singleton subsets hyperplanes so each sets consisting of a point.
• In two-dimensional space, the straight lines are the hyperplanes. This statement is for a nichtdesarguessche affine or projective plane, the definition for the term hyperplane, as the dimension and sub-space concept of linear algebra for desarguesche rooms can not be generalized to the coordinate ranges of these levels without further notice.
• In three-dimensional space, each plane is a hyperplane.

Applications

Classical geometry

• A hyperplane in at least two-dimensional Euclidean space divides the space into two half-spaces,
• This layout can be generalized pagination for affine spaces arranged over bodies with the concept of (strong),
• For (also nichtdesarguessche ) affine planes a (weak) pagination exists in certain cases by straight lines.

Finite geometry

In finite geometry have among the finite affine or projective geometries, those specific properties, which is - as the set of points in addition to the ordinary "dots" - specifically, the selected hyperplanes amount of space as a block. → Refer to the block diagram.