nLab F-norm

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

Definition

If

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

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

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

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

Definition

If

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

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

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

Definition

If

1. ${\|x + y\|} \leq {\|x\|} + {\|y\|}$ for each vector $x$ and vector $y$ and
2. ${\|a x\|} = {|a|} {\|x\|}$ for each scalar $a$ and vector $x$,

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

Every seminorm is automatically an F-seminorm.

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

Definition

If

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

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. $x$ converges under the metric on $V$ whenever $x$ is a net of vectors and $\lim_{i,j} {\|x_j - x_i\|} = 0$ in $\mathbb{R}$,

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

Definition

If

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

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

Finally, $F Sp$ 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 $l^p$ for $p \lt 0 \lt 1$. These use a modified $p$-norm in which

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

(so without the $p$th 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