nLab
F-norm

F-norms

Idea

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.

Definitions

Let KK 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 KK scalars. Let VV be a vector space (or module) over KK; we will call the elements of VV vectors. Let \|{-}\| be a function from (the underlying set of) VV to the set of real numbers.

Definition

If

  1. 0 V=0{\|0_V\|} = 0 (or even just 00{\|0\|} \leq 0),
  2. x=x{\|{-x}\|} = {\|x\|} (or even just xx{\|{-x}\|} \leq {\|x\|}) for each vector xx, and
  3. x+yx+y{\|x + y\|} \leq {\|x\|} + {\|y\|} for each vector xx and vector yy (the triangle inequality),

then {\|{-}\|} is a G-seminorm.

This is enough to prove that x0{\|x\|} \geq 0 for each xx in VV, making (x,y)yx(x,y) \mapsto {\|y - x\|} (precisely) a translation-invariant pseudometric on VV.

Note that addition (x,y)x+y:V×VV(x,y) \mapsto x + y\colon V \times V \to V is a short map under this pseudometric and so certainly continuous.

Definition

If

  1. \|{-}\| is a G-seminorm and
  2. scalar multiplication (a,x)ax:K×VV(a,x) \mapsto a x\colon K \times V \to V is continuous (relative to the topology on KK and the pseudometric on VV),

then \|{-}\| is an F-seminorm.

If the topology on KK is given by an absolute value |||{-}|, then we can go further:

Definition

If

  1. x+yx+y{\|x + y\|} \leq {\|x\|} + {\|y\|} for each vector xx and vector yy and
  2. ax=|a|x{\|a x\|} = {|a|} {\|x\|} for each scalar aa and vector xx,

then \|{-}\| is a seminorm.

Every seminorm is automatically an F-seminorm.

No longer assuming anything further about KK, there are some subsidiary definitions:

Definition

If

  1. \|{-}\| is an F-seminorm and
  2. x=0 Vx = 0_V whenever xx is a vector and x=0{\|x\|} = 0,

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.)

Definition

If

  1. \|{-}\| is an F-norm and
  2. xx converges under the metric on VV whenever xx is a net of vectors and lim i,jx jx i=0\lim_{i,j} {\|x_j - x_i\|} = 0 in \mathbb{R},

then (V,)(V,{\|{-}\|}) 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 VV is complete?).

Definition

If

  1. (V,)(V,{\|{-}\|}) is an F-space and
  2. (V,)(V,{\|{-}\|}) is locally convex as a topological vector space,

then (V,)(V,{\|{-}\|}) is a Fréchet space.

Finally, FSpF Sp 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.)

Examples

The usual examples of F-spaces that are not Fréchet spaces are the Lebesgue spaces l pl^p for p<0<1p \lt 0 \lt 1. These use a modified pp-norm in which

x p= i|x i| p {\|x\|_p} = \sum_i {|x_i|^p}

(so without the ppth root) to ensure that the triangle inequality (.3) holds.

Properties

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?.

category: analysis

Last revised on April 11, 2017 at 08:33:44. See the history of this page for a list of all contributions to it.