View source for Fréchet topology

The [[Topological structure (topology)|topological structure (topology)]] of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101802.png" />-space (a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101803.png" />; cf. also [[Fréchet space|Fréchet space]]), i.e. a completely metrizable [[Topological vector space|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101804.png" />-space if and only if it has a countable basis of neighbourhoods of the origin. The topology of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101805.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101806.png" /> can be given by means of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101808.png" />-norm, i.e. a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f1101809.png" /> satisfying:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018011.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018012.png" />;

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018013.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018014.png" />;

iii) for each scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018016.png" />, and for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018018.png" />. This means that the (complete) topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018019.png" /> can be given by means of a distance of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018020.png" />. The completion of any metrizable topological vector space (cf. [[Completion|Completion]]) is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018021.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018022.png" /> depends only upon <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018023.png" /> and that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018024.png" /> is non-decreasing for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018025.png" />. If one relaxes condition i) so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018026.png" /> can hold for a non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018027.png" />, one obtains an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018029.png" />-semi-norm. The topology of an arbitrary topological vector space can be given by means of a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018030.png" />-semi-norms; consequently, every complete topological vector space is an inverse (projective) limit of a directed family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018031.png" />-spaces.

==Important classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018032.png" />-spaces.==


===Locally convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018033.png" />-spaces.===
Such spaces are also called spaces of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018036.png" /> (some authors call them just Fréchet spaces, but see [[Fréchet space|Fréchet space]]). The topology of such a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018037.png" /> can be given by means of an increasing sequence of (homogeneous) semi-norms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

so that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018039.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018040.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018041.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018042.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018043.png" />. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018044.png" />-norm giving this topology can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018045.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018046.png" /> is a continuous [[Linear operator|linear operator]] from a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018047.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018048.png" /> to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018049.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018050.png" />, then for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018051.png" /> there are a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018052.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018053.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018055.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018056.png" /> (it is important here that the systems of semi-norms giving, respectively, the topologies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018058.png" /> satisfy (a1)). The dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018059.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018060.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018061.png" /> (the space of all continuous linear functionals provided with the topology of uniform convergence on bounded sets) is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018063.png" />-space; it is non-metrizable (unless <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018064.png" /> is a [[Banach space|Banach space]]). Any space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018065.png" /> is an inverse (projective) limit of a sequence of Banach spaces.

===Complete locally bounded spaces.===
A topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018066.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018067.png" /> is metrizable and can be given by means of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018069.png" />-homogeneous norm, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018070.png" />, i.e. an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018071.png" />-norm satisfying instead of iii) the more restrictive condition

iiia) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018072.png" /> for all scalars <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018073.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018074.png" />.

For this reason, locally bounded spaces are sometimes called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018076.png" />-normed spaces. The class of all Banach spaces is exactly the intersection of the class of locally convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018077.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018078.png" />-spaces.===
They are like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018079.png" />-spaces, but with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018080.png" />-homogeneous semi-norms instead of homogeneous semi-norms (the exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018081.png" /> may depend upon the semi-norm). This class contains the class of locally convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018082.png" />-spaces and the class of complete locally bounded spaces.

==Examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018083.png" />-spaces.==
The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018084.png" /> of all Lebesgue-measurable functions on the unit interval with the topology of convergence in measure (asymptotic convergence) is a space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018085.png" />. Its topology can be given by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018086.png" />-norm

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018087.png" /></td> </tr></table>

This space is not locally pseudo-convex.

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018088.png" /> of all infinitely differentiable functions on the unit interval with the topology of unform convergence of functions together with all derivatives is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018089.png" />-space. Its topology can be given by semi-norms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018090.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018091.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018092.png" /> of all entire functions of one complex variable with the topology of uniform convergence on compact subsets of the complex plane is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018093.png" />-space. Its topology can be given by the semi-norms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018094.png" /></td> </tr></table>

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018095.png" /> on the unit interval, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018096.png" />, is a complete locally bounded space with trivial dual. Its topology can be given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018097.png" /> (its discrete analogue, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018098.png" /> of all sequences summable with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018099.png" />-th power, has a non-trivial dual).

The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180100.png" /> with the semi-norms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180101.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180102.png" />, is a locally pseudo-convex space of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180103.png" /> which is not locally bounded.

==General facts about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180104.png" />-spaces.==
A [[Linear operator|linear operator]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180105.png" /> spaces is continuous if and only if it maps bounded sets onto bounded sets.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180106.png" /> be a family of continuous linear operators from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180107.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180108.png" /> to an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180109.png" />-space space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180110.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180111.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180112.png" /> for each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180113.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180114.png" /> is equicontinuous (the [[Mazur–Orlicz theorem|Mazur–Orlicz theorem]]; it is a theorem of Banach–Steinhaus type, cf. also [[Banach–Steinhaus theorem|Banach–Steinhaus theorem]]).

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180116.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180117.png" />-spaces and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180118.png" /> is a sequence of continuous linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180119.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180120.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180121.png" /> the limit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180122.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180123.png" /> is a continuous linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180124.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180125.png" />.

The image of an open set under a continuous linear operator between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180126.png" />-spaces is open (the open mapping theorem).

The graph of a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180127.png" /> between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180128.png" />-spaces is closed if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180129.png" /> is continuous (the closed graph theorem).

If a one-to-one continuous linear operator maps an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180130.png" />-space onto an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180131.png" />-space, then the inverse operator is continuous (the inverse operator theorem).

A separately continuous bilinear mapping between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180132.png" />-spaces is jointly continuous (cf. also [[Continuous function|Continuous function]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Banach,   "Théorie des operations lineaires" , Warszawa  (1932)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki,   "Espaces vectorielles topologiques" , Paris  (1981)  pp. Chapt. 1–5</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators" , '''I. General theory''' , Wiley, reprint  (1988)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Grothendieck,   "Topological vector spaces" , New York  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Jarchow,   "Locally convex spaces" , Teubner  (1981)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''I–II''' , New York  (1969–1979)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Rolewicz,   "Metric linear spaces" , PWN &amp; Reidel  (1972)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Springer  (1971)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  L. Waelbroeck,   "Topological vector spaces and algebras" , ''Lecture Notes in Mathematics'' , '''230''' , Springer  (1971)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  A. Wilansky,   "Modern methods in topological vector spaces" , New York  (1978)</TD></TR></table>