nLab F-space

This article is about metrizable complete topological vector spaces. For the locally convex special case, see the article Fréchet space.

Terminology

There are two inequivalent definitions of Fréchet spaces found in the literature. The original definition due to Stefan Banach defines Fréchet spaces as metrizable complete topological vector spaces.

Later Bourbaki (Topological vector spaces, Section II.4.1) added the condition of local convexity. However, many authors continue to use the original definition due to Banach.

The term “F-space” can refer to either of these definitions, although in the modern literature it is more commonly used to refer to the non-locally convex notion.

The nLab uses “F-space” to refer to the non-locally convex notion and “Fréchet space” to refer to the locally convex notion.

Idea

An F-space is a topological vector space whose topology is induced by a complete F-norm.

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 $0 \lt p \lt 1$. 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 induce the same topology. See norm#dreamUnique.

Related concepts

• Fréchet space

References

• Nigel Kalton, Quasi-Banach spaces, Handbook of the geometry of Banach spaces, Volume 2, 1099–1130. North-Holland, Amsterdam, 2003. ISBN: 0-444-51305-1.

• N. Kalton, N. T. Peck, James W. Roberts, An F-space sampler, Cambridge University Press (1984), London Mathematical Society Lecture Notes 89, Cambridge. ISBN: 9780511662447, DOI.

• Norbert Adasch, Bruno Ernst, Dieter Keim, Topological Vector Spaces: The Theory Without Convexity Conditions, Lecture Notes in Mathematics 639 (1978), Springer. ISBN 978-3-540-08662-8.

category: analysis