Minkowski functional

In the mathematical branch of functional analysis, the Minkowski functional is (after Hermann Minkowski ), often called functional verification, a generalization of the standard notion.

Definition

It is a topological vector space. Is now an absorbing subset, so the function is called

The Minkowski functional or functional verification to.

Properties

• Is balanced and convex, so is a semi-norm or seminorm. Conversely, the above-mentioned properties for each semi standard amount. It follows that the locally convex spaces are precisely the spaces whose topology can be defined by a separating family of semi- norms.

• Is poised limited and convex, the Minkowski functional is a norm that induces the given topology. In particular, a topological vector space if and only normalizable if there is a bounded convex neighborhood of the root.

Example

In a Euclidean space (such as the three-dimensional space of everyday intuition ), consider as a subset of the unit sphere. Then the Minkowski functional is identical to the usual Euclidean norm, because with lies just on the edge of the crowd, so the sphere with radius and center 0