Fréchet topology

The topological structure (topology) of an $F$- space (a space of type $F$; cf. also Fréchet space), i.e. a completely metrizable topological vector space. The term was introduced by S. Banach in honour of M. Fréchet. Many authors, however, demand additionally local convexity. A complete topological vector space is an $F$- space if and only if it has a countable basis of neighbourhoods of the origin. The topology of an $F$- space $X$ can be given by means of an $F$- norm, i.e. a function $x \mapsto \| x \|$ satisfying:

i) $\| x \| \geq 0$ and $\| x \| = 0$ if and only if $x = 0$;

ii) $\| {x + y } \| \leq \| x \| + \| y \|$ for all $x,y \in X$;

iii) for each scalar $\lambda$, ${\lim\limits } _ {\| x \| \rightarrow 0 } \| {\lambda x } \| = 0$, and for each $x \in X$, ${\lim\limits } _ {| \lambda | \rightarrow 0 } \| {\lambda x } \| = 0$. This means that the (complete) topology of $X$ can be given by means of a distance of the form $d ( x,y ) = \| {x - y } \|$. The completion of any metrizable topological vector space (cf. Completion) is an $F$- space and, consequently, the topology of any metric vector space can be given by means of a translation-invariant distance. Without loss of generality it can be assumed that $\| {\lambda x } \|$ depends only upon $| \lambda |$ and that the function $| \lambda | \mapsto \| {\lambda x } \|$ is non-decreasing for each $x \in X$. If one relaxes condition i) so that $\| x \| = 0$ can hold for a non-zero $x$, one obtains an $F$- semi-norm. The topology of an arbitrary topological vector space can be given by means of a family of $F$- semi-norms; consequently, every complete topological vector space is an inverse (projective) limit of a directed family of $F$- spaces.

Important classes of $F$-spaces.

Locally convex $F$-spaces.

Such spaces are also called spaces of type $B _ {o}$( some authors call them just Fréchet spaces, but see Fréchet space). The topology of such a space $X$ can be given by means of an increasing sequence of (homogeneous) semi-norms

$$\tag{a1 } \left \| x \right \| _ {1} \leq \left \| x \right \| _ {2} \leq \dots , \textrm{ for all } x \in X,$$

so that a sequence $( x _ {k} )$ of elements of $X$ tends to $0$ if and only if ${\lim\limits } _ {k} \| {x _ {k} } \| _ {n} = 0$ for $n = 1,2, \dots$. An $F$- norm giving this topology can be written as

$$\left \| x \right \| = \sum _ { 1 } ^ \infty 2 ^ {- n } { \frac{\left \| x \right \| _ {n} }{1 + \left \| x \right \| _ {n} } } .$$

If $T$ is a continuous linear operator from a $B _ {o}$- space $X$ to a $B _ {o}$- space $Y$, then for each $n$ there are a $k ( n )$ and a $C _ {n} > 0$ such that $\| {Tx } \| _ {n} ^ {( Y ) } \leq C _ {n} \| x \| _ {k ( n ) } ^ {( X ) }$, $x \in X$, for all $n$( it is important here that the systems of semi-norms giving, respectively, the topologies of $X$ and $Y$ satisfy (a1)). The dual space $X ^ \prime$ of a $B _ {o}$- space $X$( the space of all continuous linear functionals provided with the topology of uniform convergence on bounded sets) is said to be an $LF$- space; it is non-metrizable (unless $X$ is a Banach space). Any space of type $B _ {o}$ is an inverse (projective) limit of a sequence of Banach spaces.

Complete locally bounded spaces.

A topological vector space $X$ is said to be locally bounded if it has a bounded neighbourhood of the origin (then it has a basis of such neighbourhoods consisting of bounded sets). The topology of such a space $X$ is metrizable and can be given by means of a $p$- homogeneous norm, $0 < p \leq 1$, i.e. an $F$- norm satisfying instead of iii) the more restrictive condition

iiia) $\| {\lambda x } \| = | \lambda | ^ {p} \| x \|$ for all scalars $\lambda$ and all $x \in X$.

For this reason, locally bounded spaces are sometimes called $p$- normed spaces. The class of all Banach spaces is exactly the intersection of the class of locally convex $F$- spaces and the class of complete locally bounded spaces. The dual space of a locally bounded space can be trivial, i.e. consist only of a zero functional.

Locally pseudo-convex $F$-spaces.

They are like $B _ {o}$- spaces, but with $p$- homogeneous semi-norms instead of homogeneous semi-norms (the exponent $p$ may depend upon the semi-norm). This class contains the class of locally convex $F$- spaces and the class of complete locally bounded spaces.

Examples of $F$-spaces.

The space $S [ 0,1 ]$ of all Lebesgue-measurable functions on the unit interval with the topology of convergence in measure (asymptotic convergence) is a space of type $F$. Its topology can be given by the $F$- norm

$$\left \| x \right \| = \int\limits _ { 0 } ^ { 1 } { { \frac{\left | {x ( t ) } \right | }{1 + \left | {x ( t ) } \right | } } } {dt } .$$

This space is not locally pseudo-convex.

The space $C ^ \infty [ 0,1 ]$ of all infinitely differentiable functions on the unit interval with the topology of unform convergence of functions together with all derivatives is a $B _ {o}$- space. Its topology can be given by semi-norms

$$\left \| x \right \| _ {n} =$$

$$= \max \left \{ \max _ {[ 0,1 ] } \left | {x ( t ) } \right | , \max _ {[ 0,1 ] } \left | {x ^ \prime ( t ) } \right | \dots \max _ {[ 0,1 ] } \left | {x ^ {( n - 1 ) } ( t ) } \right | \right \} .$$

The space ${\mathcal E}$ of all entire functions of one complex variable with the topology of uniform convergence on compact subsets of the complex plane is a $B _ {o}$- space. Its topology can be given by the semi-norms

$$\left \| x \right \| _ {n} = \max _ {\left | \zeta \right | \leq n } \left | {x ( \zeta ) } \right | .$$

The space $L _ {p} [ 0,1 ]$ on the unit interval, $0 < p < 1$, is a complete locally bounded space with trivial dual. Its topology can be given by $\| x \| _ {p} = \int _ {0} ^ {1} {| {x ( t ) } | ^ {p} } {dt }$( its discrete analogue, the space $l _ {p}$ of all sequences summable with the $p$- th power, has a non-trivial dual).

The space $L ^ {0 + } [ 0,1 ] = \cap _ {0 < p \leq 1 } L _ {p} [ 0,1 ]$ with the semi-norms $\| x \| _ {p _ {n} }$, where $0 < p _ {n} \rightarrow 0$, is a locally pseudo-convex space of type $F$ which is not locally bounded.

General facts about $F$-spaces.

A linear operator between $F$ spaces is continuous if and only if it maps bounded sets onto bounded sets.

Let ${\mathcal A}$ be a family of continuous linear operators from an $F$- space $X$ to an $F$- space space $Y$. If the set $\{ {Tx } : {T \in {\mathcal A} } \}$ is bounded in $Y$ for each fixed $x \in X$, then ${\mathcal A}$ is equicontinuous (the Mazur–Orlicz theorem; it is a theorem of Banach–Steinhaus type, cf. also Banach–Steinhaus theorem).

If $X$ and $Y$ are $F$- spaces and $( T _ {n} )$ is a sequence of continuous linear operators from $X$ to $Y$ such that for each $x \in X$ the limit $Tx = {\lim\limits } _ {n} T _ {n} x$ exists, then $x \mapsto Tx$ is a continuous linear operator from $X$ to $Y$.

The image of an open set under a continuous linear operator between $F$- spaces is open (the open mapping theorem).

The graph of a linear operator $T$ between $F$- spaces is closed if and only if $T$ is continuous (the closed graph theorem).

If a one-to-one continuous linear operator maps an $F$- space onto an $F$- space, then the inverse operator is continuous (the inverse operator theorem).

A separately continuous bilinear mapping between $F$- spaces is jointly continuous (cf. also Continuous function).

References

[a1]	S. Banach, "Théorie des opérations linéaires" , Warszawa (1932)
[a2]	N. Bourbaki, "Espaces vectoriels topologiques" , Paris (1981) pp. Chapt. 1–5
[a3]	N. Dunford, J.T. Schwartz, "Linear operators" , I. General theory , Wiley, reprint (1988)
[a4]	A. Grothendieck, "Topological vector spaces" , New York (1973)
[a5]	H. Jarchow, "Locally convex spaces" , Teubner (1981)
[a6]	G. Köthe, "Topological vector spaces" , I–II , New York (1969–1979)
[a7]	S. Rolewicz, "Metric linear spaces" , PWN & Reidel (1972)
[a8]	H.H. Schaefer, "Topological vector spaces" , Springer (1971)
[a9]	L. Waelbroeck, "Topological vector spaces and algebras" , Lecture Notes in Mathematics , 230 , Springer (1971)
[a10]	A. Wilansky, "Modern methods in topological vector spaces" , New York (1978)