An F-norm is a non-homogeneous variant of a norm: a translation-invariant metric on a vector space that satisfies properties in between being a G-norm (on the underlying abelian group of the vector space) and being a norm. As with norms, there is a semi- variant.
Let be a topological field (typically the real numbers or the complex numbers, but conceivably only a topological ring, or at least a commutative one); we will call the elements of scalars. Let be a vector space (or module) over ; we will call the elements of vectors. Let be a function from (the underlying set of) to the set of real numbers.
If
then is a G-seminorm.
This is enough to prove that for each in , making (precisely) a translation-invariant pseudometric on .
Note that addition is a short map under this pseudometric and so certainly continuous.
If
then is an F-seminorm.
If the topology on is given by an absolute value , then we can go further:
Every seminorm is automatically an F-seminorm.
No longer assuming anything further about , there are some subsidiary definitions:
If
then is an F-norm.
Thus an F-norm is precisely an F-seminorm whose induced pseudometric is a metric. (Compare the relationship between G-norms and norms with G-seminorms in and seminorms in above.)
If
then is an F-space.
In other words, an F-space is a vector space equipped with an F-norm whose induced metric is complete (or equivalently such that the topology on is complete?).
If
then is a Fréchet space.
Finally, is the category whose objects are F-spaces and whose morphisms are short linear maps; that said, often people really study the essential image of that category within the category of topological vector spaces, or equivalently the category whose objects are F-spaces and whose morphisms are continuous linear maps. (This is especially so with Fréchet spaces, which have a common alternative definition that makes no reference to a canonical metric.)
The usual examples of F-spaces that are not Fréchet spaces are the Lebesgue spaces for . These use a modified -norm in which
(so without the th root) to ensure that the triangle inequality (.3) holds.
The uniqueness theorem for complete norms in dream mathematics applies also to F-norms: assuming excluded middle, dependent choice, and the (classically false) Borel property?, two complete F-norms on a given vector space over the real numbers must be topologically equivalent?. See norm#dreamUnique.
Last revised on April 11, 2017 at 12:33:44. See the history of this page for a list of all contributions to it.