Science Done Right

Minkowski Functionals and Seminorms

3 min read

Minkowski Functional

Suppose is a subset of a real or complex vector space. Then the Minkowski functional or gauge of is defined to be a function such that

Conventionally, we set the infimum over an empty set to be .

e.g.

  • Consider a normed vector space and let be the unit ball in . Then the Minkowski functional is just the norm on .

Seminorms

In practice, it is often useful to define a locally convex space by means of a family of seminorms.

Seminorm

A seminorm on a complex or real topological vector space is a map such that the following properties hold:

  • nonnegativity: for all ,
  • absolute homogeneity: for all and ,
  • triangle inequality: for all .

Alternative Characterization of Locally Convex Spaces

A topological vector space is locally convex if and only if its topology is induced by a countable family of seminorms. That is, its topology is the coarsest topology such that each seminorm is continuous.

Fréchet Spaces

Lemma

A complete topological vector space is complete in every compatible metric that induces its topology.

Fréchet Space

A topological vector space is a Fréchet space if it is, locally convex, metrizable and complete.

Remark

The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces, which does not require local convexity. The books of Waelbroeck (1971) and Wilansky (1978) follow this original definition.

The following propositions help to verify if a given space is Fréchet:

Proposition

A locally convex space is metrizable if and only if its topology can be generated by a countable separating (for every non-zero , there must be at least one seminorm ​ such that ) family of seminorms.

e.g.

  • Clearly, any Banach space is a Fréchet space, since it is complete and metrizable by the norm.
  • The space of real smooth functions is a Fréchet space with the seminorms defined by for each nonnegative integers and .
  • If is a compact smooth manifold, then the space of smooth functions is a Fréchet space. This holds for some non-compact manifolds as well, such as the real line .

Graph View