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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:
+
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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" />;
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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" />;
+
The [[Topological structure (topology)|topological structure (topology)]] of an  $  F $-
 +
space (a space of type  $  F $;
 +
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  $  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:
  
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.
+
i) $  \| x \| \geq  0 $
 +
and $  \| x \| = 0 $
 +
if and only if  $  x = 0 $;
  
==Important classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018032.png" />-spaces.==
+
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|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.
  
===Locally convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018033.png" />-spaces.===
+
==Important classes of  $  F $-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>
+
===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|Fréchet space]]). The topology of such a space  $  X $
 +
can be given by means of an increasing sequence of (homogeneous) semi-norms
  
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
+
$$ \tag{a1 }
 +
\left \| x \right \| _ {1} \leq  \left \| x \right \| _ {2} \leq  \dots ,  \textrm{ for all  }  x \in X,
 +
$$
  
<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>
+
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
  
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.
+
$$
 +
\left \| x \right \| = \sum _ { 1 } ^  \infty  2 ^ {- n } {
 +
\frac{\left \| x \right \| _ {n} }{1 + \left \| x \right \| _ {n} }
 +
} .
 +
$$
 +
 
 +
If  $  T $
 +
is a continuous [[Linear operator|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|Banach space]]). Any space of type $  B _ {o} $
 +
is an inverse (projective) limit of a sequence of Banach spaces.
  
 
===Complete locally bounded 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
+
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) <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" />.
+
iiia) $  \| {\lambda x } \| = | \lambda |  ^ {p} \| x \| $
 +
for all scalars $  \lambda $
 +
and all $  x \in X $.
  
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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018078.png" />-spaces.===
+
===Locally pseudo-convex $  F $-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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f11018083.png" />-spaces.==
+
==Examples of $  F $-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
+
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
  
<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>
+
$$
 +
\left \| x \right \| = \int\limits _ { 0 } ^ { 1 }  { {
 +
\frac{\left | {x ( t ) } \right | }{1 + \left | {x ( t ) } \right | }
 +
} }  {dt } .
 +
$$
  
 
This space is not locally pseudo-convex.
 
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
+
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
  
<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>
+
$$
 +
\left \| x \right \| _ {n} =
 +
$$
  
<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>
+
$$
 +
=  
 +
\max  \left \{  \max  _ {[ 0,1 ] } \left | {x ( t ) } \right | ,  \max  _ {[ 0,1 ] } \left | {x  ^  \prime  ( t ) } \right | \dots  \max  _ {[ 0,1 ] } \left | {x ^ {( n - 1 ) } ( t ) } \right | \right \} .
 +
$$
  
The space <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
+
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
  
<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>
+
$$
 +
\left \| x \right \| _ {n} = \max  _ {\left | \zeta \right | \leq  n } \left | {x ( \zeta ) } \right | .
 +
$$
  
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 $  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 <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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180104.png" />-spaces.==
+
==General facts about $  F $-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.
+
A [[Linear operator|linear operator]] between $  F $
 +
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]]).
+
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|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" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110180/f110180126.png" />-spaces is open (the open mapping theorem).
+
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 <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).
+
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 <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).
+
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 <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]]).
+
A separately continuous bilinear mapping between $  F $-
 +
spaces is jointly continuous (cf. also [[Continuous function|Continuous function]]).
  
 
====References====
 
====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>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Banach, "Théorie des opérations linéaires" , Warszawa  (1932)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Bourbaki, "Espaces vectoriels 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>

Latest revision as of 18:52, 25 February 2024


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)
How to Cite This Entry:
Fréchet topology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_topology&oldid=12759
This article was adapted from an original article by W. Zelazko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article