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The topological structure (topology) of an -space (a space of type
; 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
-space if and only if it has a countable basis of neighbourhoods of the origin. The topology of an
-space
can be given by means of an
-norm, i.e. a function
satisfying:
i) and
if and only if
;
ii) for all
;
iii) for each scalar ,
, and for each
,
. This means that the (complete) topology of
can be given by means of a distance of the form
. The completion of any metrizable topological vector space (cf. Completion) is an
-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
depends only upon
and that the function
is non-decreasing for each
. If one relaxes condition i) so that
can hold for a non-zero
, one obtains an
-semi-norm. The topology of an arbitrary topological vector space can be given by means of a family of
-semi-norms; consequently, every complete topological vector space is an inverse (projective) limit of a directed family of
-spaces.
Important classes of
-spaces.
Locally convex
-spaces.
Such spaces are also called spaces of type (some authors call them just Fréchet spaces, but see Fréchet space). The topology of such a space
can be given by means of an increasing sequence of (homogeneous) semi-norms
![]() | (a1) |
so that a sequence of elements of
tends to
if and only if
for
. An
-norm giving this topology can be written as
![]() |
If is a continuous linear operator from a
-space
to a
-space
, then for each
there are a
and a
such that
,
, for all
(it is important here that the systems of semi-norms giving, respectively, the topologies of
and
satisfy (a1)). The dual space
of a
-space
(the space of all continuous linear functionals provided with the topology of uniform convergence on bounded sets) is said to be an
-space; it is non-metrizable (unless
is a Banach space). Any space of type
is an inverse (projective) limit of a sequence of Banach spaces.
Complete locally bounded spaces.
A topological vector space 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
is metrizable and can be given by means of a
-homogeneous norm,
, i.e. an
-norm satisfying instead of iii) the more restrictive condition
iiia) for all scalars
and all
.
For this reason, locally bounded spaces are sometimes called -normed spaces. The class of all Banach spaces is exactly the intersection of the class of locally convex
-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
-spaces.
They are like -spaces, but with
-homogeneous semi-norms instead of homogeneous semi-norms (the exponent
may depend upon the semi-norm). This class contains the class of locally convex
-spaces and the class of complete locally bounded spaces.
Examples of
-spaces.
The space of all Lebesgue-measurable functions on the unit interval with the topology of convergence in measure (asymptotic convergence) is a space of type
. Its topology can be given by the
-norm
![]() |
This space is not locally pseudo-convex.
The space of all infinitely differentiable functions on the unit interval with the topology of unform convergence of functions together with all derivatives is a
-space. Its topology can be given by semi-norms
![]() |
![]() |
The space of all entire functions of one complex variable with the topology of uniform convergence on compact subsets of the complex plane is a
-space. Its topology can be given by the semi-norms
![]() |
The space on the unit interval,
, is a complete locally bounded space with trivial dual. Its topology can be given by
(its discrete analogue, the space
of all sequences summable with the
-th power, has a non-trivial dual).
The space with the semi-norms
, where
, is a locally pseudo-convex space of type
which is not locally bounded.
General facts about
-spaces.
A linear operator between spaces is continuous if and only if it maps bounded sets onto bounded sets.
Let be a family of continuous linear operators from an
-space
to an
-space space
. If the set
is bounded in
for each fixed
, then
is equicontinuous (the Mazur–Orlicz theorem; it is a theorem of Banach–Steinhaus type, cf. also Banach–Steinhaus theorem).
If and
are
-spaces and
is a sequence of continuous linear operators from
to
such that for each
the limit
exists, then
is a continuous linear operator from
to
.
The image of an open set under a continuous linear operator between -spaces is open (the open mapping theorem).
The graph of a linear operator between
-spaces is closed if and only if
is continuous (the closed graph theorem).
If a one-to-one continuous linear operator maps an -space onto an
-space, then the inverse operator is continuous (the inverse operator theorem).
A separately continuous bilinear mapping between -spaces is jointly continuous (cf. also Continuous function).
References
[a1] | S. Banach, "Théorie des operations lineaires" , Warszawa (1932) |
[a2] | N. Bourbaki, "Espaces vectorielles 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) |
Fréchet topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_topology&oldid=22463