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A [[Topological space|topological space]] containing a homeomorphic image of every topological space of a certain class. Examples are: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957601.png" />, cf. [[Banach space|Banach space]]; 2) the [[Hilbert cube|Hilbert cube]] and the [[Tikhonov cube|Tikhonov cube]]; 3) the Menger curve (cf. [[Line (curve)|Line (curve)]]); 4) the universal Milnor bundle (cf. [[Principal fibre bundle|Principal fibre bundle]]).
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A [[Topological space|topological space]] containing a homeomorphic image of every topological space of a certain class. Examples are: 1) $  C [ 0 , 1 ] $,
 +
cf. [[Banach space|Banach space]]; 2) the [[Hilbert cube|Hilbert cube]] and the [[Tikhonov cube|Tikhonov cube]]; 3) the Menger curve (cf. [[Line (curve)|Line (curve)]]); 4) the universal Milnor bundle (cf. [[Principal fibre bundle|Principal fibre bundle]]).
  
 
The existence of universal spaces allows the consideration of abstract objects as subobjects (in the categorical sense) of a more concrete one, and thus endows them with a greater wealth of  "intrinsic"  properties. On the other hand, it emphasizes the relations of  "parts of a whole" .
 
The existence of universal spaces allows the consideration of abstract objects as subobjects (in the categorical sense) of a more concrete one, and thus endows them with a greater wealth of  "intrinsic"  properties. On the other hand, it emphasizes the relations of  "parts of a whole" .
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,  "Universal algebra" , Reidel  (1981)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
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==Universal spaces in functional analysis.==
 
==Universal spaces in functional analysis.==
There are various notions of a universal space in functional analysis. A topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957602.png" /> is universal for a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957603.png" /> of topological vector spaces if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957604.png" /> there is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957605.png" /> isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957606.png" />. There is always a trivial universal space for any class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957607.png" />, but whether there is a universal space in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957608.png" /> itself or in a closely related class is a different matter. The following theorem holds, [[#References|[a2]]]: There is a universal separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u0957609.png" />-space for the class of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576010.png" />-spaces. Here, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576011.png" />-space and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576012.png" />-space are defined as follows.
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There are various notions of a universal space in functional analysis. A topological vector space $  X _ {0} $
 +
is universal for a class $  {\mathcal X} $
 +
of topological vector spaces if for every $  X \in {\mathcal X} $
 +
there is a closed subspace of $  X _ {0} $
 +
isomorphic to $  X $.  
 +
There is always a trivial universal space for any class $  {\mathcal X} $,  
 +
but whether there is a universal space in $  {\mathcal X} $
 +
itself or in a closely related class is a different matter. The following theorem holds, [[#References|[a2]]]: There is a universal separable $  F $-space for the class of all $  F ^ { * } $-spaces. Here, an $  F $-space and an $  F ^ { * } $-space are defined as follows.
  
An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576014.png" />-norm on a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576015.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576016.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576017.png" /> to the non-negative real numbers such that:
+
An $  F $-norm on a linear space $  X $
 +
is a mapping $  \| \| $
 +
from $  X $
 +
to the non-negative real numbers such that:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576018.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576019.png" />;
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1) $  \| x \| = 0 $
 +
if and only if $  x= 0 $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576020.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576022.png" />;
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2) $  \| ax \| = \| x \| $
 +
for all $  a $,  
 +
$  | a | = 1 $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576023.png" />;
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3) $  \| x+ y \| \leq  \| x \| + \| y \| $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576025.png" />;
+
4) $  \| a _ {n} x \| \rightarrow 0 $
 +
if $  a _ {n} \rightarrow 0 $;
  
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576027.png" />;
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5) $  \| ax _ {n} \| \rightarrow 0 $
 +
if $  x _ {n} \rightarrow 0 $;
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576030.png" />.
+
6) $  \| a _ {n} x _ {n} \| \rightarrow 0 $
 +
if $  a _ {n} \rightarrow 0 $,  
 +
$  x _ {n} \rightarrow 0 $.
  
In conditions 4), 5), 6) the topology is the one induced by the translation-invariant metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576031.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576032.png" /> is not required to satisfy 1), it is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576034.png" />-pseudo-norm. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576035.png" />-norm or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576036.png" />-pseudo-norm is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576039.png" />-homogeneous if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576040.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576043.png" />-homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576044.png" />-norm (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576045.png" />-pseudo-norm) is a [[Norm|norm]] ([[Pseudo-norm|pseudo-norm]]). An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576047.png" />-space is a linear space with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576048.png" />-norm; an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576050.png" />-space is a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576051.png" />-space.
+
In conditions 4), 5), 6) the topology is the one induced by the translation-invariant metric $  \rho ( x, y)= \| x- y \| $.  
 +
If $  \| \| $
 +
is not required to satisfy 1), it is called an $  F $-pseudo-norm. An $  F $-norm or $  F $-pseudo-norm is $  p $-homogeneous if $  \| ax \| = | a |  ^ {p} \| x \| $.  
 +
A $  1 $-homogeneous $  F $-norm ( $  F $-pseudo-norm) is a [[Norm|norm]] ([[Pseudo-norm|pseudo-norm]]). An $  F ^ { * } $-space is a linear space with an $  F $-norm; an $  F $-space is a complete $  F ^ { * } $-space.
  
Some other universality results: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576052.png" /> is universal for all separable Banach spaces (the Banach–Mazur theorem, cf. [[Metric space|Metric space]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576053.png" /> is universal for all separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576054.png" />-spaces (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576056.png" />-space is a locally convex metric linear space and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576058.png" />-space is a complete <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576059.png" />-space); there is a separable locally pseudo-convex space which is universal for all separable locally pseudo-convex spaces; there is a separable locally bounded complete space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576060.png" /> with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576061.png" />-homogeneous norm which is universal for all separable locally bounded spaces with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576062.png" />-homogeneous norms. (Here, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576063.png" />-space is locally bounded if it contains a bounded neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576064.png" />; a locally pseudo-convex space is a metric linear space whose topology can be given by a family of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576065.png" />-homogeneous pseudo-norms.)
+
Some other universality results: $  C[ 0, 1] $
 +
is universal for all separable Banach spaces (the Banach–Mazur theorem, cf. [[Metric space|Metric space]]); $  C( - \infty , \infty ) $
 +
is universal for all separable $  B _ {0} $-spaces (a $  B _ {0} ^ { * } $-space is a locally convex metric linear space and a $  B _ {0} $-space is a complete $  B _ {0} ^ { * } $-space); there is a separable locally pseudo-convex space which is universal for all separable locally pseudo-convex spaces; there is a separable locally bounded complete space $  X $
 +
with a $  p $-homogeneous norm which is universal for all separable locally bounded spaces with $  p $-homogeneous norms. (Here, an $  F $-space is locally bounded if it contains a bounded neighbourhood of 0 $;  
 +
a locally pseudo-convex space is a metric linear space whose topology can be given by a family of $  p _ {n} $-homogeneous pseudo-norms.)
  
The dual notion is that of a co-universal linear space. An <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576066.png" />-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576067.png" /> is co-universal for a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576068.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576069.png" />-spaces if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576070.png" /> is isomorphic to a quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576071.png" /> for a closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576072.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576073.png" />.
+
The dual notion is that of a co-universal linear space. An $  F $-space $  X _ {0} $
 +
is co-universal for a family $  {\mathcal X} $
 +
of $  F $-spaces if every element of $  {\mathcal X} $
 +
is isomorphic to a quotient space $  X _ {0} /Y $
 +
for a closed subspace $  Y $
 +
of $  X _ {0} $.
  
Some co-universality results: there is a separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576074.png" />-space which is co-universal for all separable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576075.png" />-spaces, [[#References|[a2]]]; every separable locally bounded space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576076.png" /> with a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576077.png" />-homogeneous norm is an image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576078.png" /> under a continuous linear operator, [[#References|[a3]]]–[[#References|[a5]]].
+
Some co-universality results: there is a separable $  F $-space which is co-universal for all separable $  F $-spaces, [[#References|[a2]]]; every separable locally bounded space $  X $
 +
with a $  p $-homogeneous norm is an image of $  l _ {p} $
 +
under a continuous linear operator, [[#References|[a3]]]–[[#References|[a5]]].
  
 
Cf. [[#References|[a6]]] for a large number of universality and co-universality theorems for metric linear spaces, including all of the above.
 
Cf. [[#References|[a6]]] for a large number of universality and co-universality theorems for metric linear spaces, including all of the above.

Latest revision as of 07:39, 14 June 2022


A topological space containing a homeomorphic image of every topological space of a certain class. Examples are: 1) $ C [ 0 , 1 ] $, cf. Banach space; 2) the Hilbert cube and the Tikhonov cube; 3) the Menger curve (cf. Line (curve)); 4) the universal Milnor bundle (cf. Principal fibre bundle).

The existence of universal spaces allows the consideration of abstract objects as subobjects (in the categorical sense) of a more concrete one, and thus endows them with a greater wealth of "intrinsic" properties. On the other hand, it emphasizes the relations of "parts of a whole" .

References

[1] P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] P.M. Cohn, "Universal algebra" , Reidel (1981)

Comments

There is also a dual notion of universal space: Every space in a certain class is a continuous image of the space in question. E.g., the Cantor set is universal for the class of compact metric spaces, the unit interval is universal for the class of locally connected continua (see Peano curve), and the pseudo-arc is universal for the class of snake-like continua (see Cube-like continuum).

Universal spaces in functional analysis.

There are various notions of a universal space in functional analysis. A topological vector space $ X _ {0} $ is universal for a class $ {\mathcal X} $ of topological vector spaces if for every $ X \in {\mathcal X} $ there is a closed subspace of $ X _ {0} $ isomorphic to $ X $. There is always a trivial universal space for any class $ {\mathcal X} $, but whether there is a universal space in $ {\mathcal X} $ itself or in a closely related class is a different matter. The following theorem holds, [a2]: There is a universal separable $ F $-space for the class of all $ F ^ { * } $-spaces. Here, an $ F $-space and an $ F ^ { * } $-space are defined as follows.

An $ F $-norm on a linear space $ X $ is a mapping $ \| \| $ from $ X $ to the non-negative real numbers such that:

1) $ \| x \| = 0 $ if and only if $ x= 0 $;

2) $ \| ax \| = \| x \| $ for all $ a $, $ | a | = 1 $;

3) $ \| x+ y \| \leq \| x \| + \| y \| $;

4) $ \| a _ {n} x \| \rightarrow 0 $ if $ a _ {n} \rightarrow 0 $;

5) $ \| ax _ {n} \| \rightarrow 0 $ if $ x _ {n} \rightarrow 0 $;

6) $ \| a _ {n} x _ {n} \| \rightarrow 0 $ if $ a _ {n} \rightarrow 0 $, $ x _ {n} \rightarrow 0 $.

In conditions 4), 5), 6) the topology is the one induced by the translation-invariant metric $ \rho ( x, y)= \| x- y \| $. If $ \| \| $ is not required to satisfy 1), it is called an $ F $-pseudo-norm. An $ F $-norm or $ F $-pseudo-norm is $ p $-homogeneous if $ \| ax \| = | a | ^ {p} \| x \| $. A $ 1 $-homogeneous $ F $-norm ( $ F $-pseudo-norm) is a norm (pseudo-norm). An $ F ^ { * } $-space is a linear space with an $ F $-norm; an $ F $-space is a complete $ F ^ { * } $-space.

Some other universality results: $ C[ 0, 1] $ is universal for all separable Banach spaces (the Banach–Mazur theorem, cf. Metric space); $ C( - \infty , \infty ) $ is universal for all separable $ B _ {0} $-spaces (a $ B _ {0} ^ { * } $-space is a locally convex metric linear space and a $ B _ {0} $-space is a complete $ B _ {0} ^ { * } $-space); there is a separable locally pseudo-convex space which is universal for all separable locally pseudo-convex spaces; there is a separable locally bounded complete space $ X $ with a $ p $-homogeneous norm which is universal for all separable locally bounded spaces with $ p $-homogeneous norms. (Here, an $ F $-space is locally bounded if it contains a bounded neighbourhood of $ 0 $; a locally pseudo-convex space is a metric linear space whose topology can be given by a family of $ p _ {n} $-homogeneous pseudo-norms.)

The dual notion is that of a co-universal linear space. An $ F $-space $ X _ {0} $ is co-universal for a family $ {\mathcal X} $ of $ F $-spaces if every element of $ {\mathcal X} $ is isomorphic to a quotient space $ X _ {0} /Y $ for a closed subspace $ Y $ of $ X _ {0} $.

Some co-universality results: there is a separable $ F $-space which is co-universal for all separable $ F $-spaces, [a2]; every separable locally bounded space $ X $ with a $ p $-homogeneous norm is an image of $ l _ {p} $ under a continuous linear operator, [a3][a5].

Cf. [a6] for a large number of universality and co-universality theorems for metric linear spaces, including all of the above.

References

[a1] R. Engelking, "General topology" , Heldermann (1989)
[a2] N.J. Kalton, "Universal spaces and universal bases in metric linear spaces" Studia Math. , 61 (1977) pp. 161–191
[a3] J.H. Shapiro, "Examples of proper closed weakly dense subspaces in non-locally convex -spaces" Isr. J. Math. , 7 (1969) pp. 369–380
[a4] W.J. Stiles, "On properties of subspaces of , " Trans. Amer. Math. Soc. , 149 (1970) pp. 405–415
[a5] S. Banach, "Théorie des opérations linéaires" , PWN (1932)
[a6] S. Rolewicz, "Metric linear spaces" , Reidel (1985) pp. 44
How to Cite This Entry:
Universal space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universal_space&oldid=18198
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article