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An operation which brings a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236901.png" /> of a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236902.png" /> into correspondence with another subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236903.png" /> so that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236905.png" /> are known, it is possible in some way to reproduce <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236906.png" />. Depending on the structure with which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236907.png" /> is endowed, one distinguishes various definitions of complementation, as well as various methods of reconstituting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236908.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c0236909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369010.png" />.
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{{TEX|part}}
  
In the general theory of sets the complement of a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369011.png" /> (or complementary subset) in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369012.png" /> is the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369013.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369015.png" />) consisting of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369016.png" /> not belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369017.png" />; an important property is the duality principle:
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An operation which brings a subset $M$ of a given set $X$ into correspondence with another subset $N$ so that if $M$ and $N$ are known, it is possible in some way to reproduce $X$. Depending on the structure with which $X$ is endowed, one distinguishes various definitions of complementation, as well as various methods of reconstituting $X$ from $M$ and $N$.
  
<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/c/c023/c023690/c02369018.png" /></td> </tr></table>
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===Sets===
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In the general theory of sets the complement of a subset $M$ (or complementary subset, relative complement) in a set $X$ is the subset $\complement_X M$ (or $\complement M$ if $X$ is assumed, or $X \setminus M$) consisting of all elements $x \in X$ not belonging to $M$; an important property is the [[duality principle]] (one of the [[De Morgan laws]]):
 +
$$
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\complement \bigcup_{\xi} M_\xi = \bigcap_\xi \complement M_\xi
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369019.png" /> have a structure of a linear space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369020.png" /> be a subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369021.png" />. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369022.png" /> is said to be a direct algebraic complement (or algebraic complement, for short) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369023.png" /> if any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369024.png" /> can be uniquely represented as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369027.png" />. This is equivalent to the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369028.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369029.png" />. Any subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369030.png" /> has an algebraic complement, but this complement is not uniquely determined.
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===Linear spaces===
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Let $X$ have a structure of a [[linear space]] and let $M$ be a subspace of $X$. A subspace $N \subset X$ is said to be a direct algebraic complement (or '''algebraic complement''', for short) of $M$ if any $x \in X$ can be uniquely represented as $x = y+z$, $y \in M$, $z \in N$. This is equivalent to the conditions $X = M + N$; $M \cap N = \{0\}$. Any subspace of $X$ has an algebraic complement, but this complement is not uniquely determined.
  
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===Linear topological spaces===
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369031.png" /> be a linear topological space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369032.png" /> be the direct algebraic sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369033.png" /> of its subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369035.png" />, regarded as linear topological spaces with the induced topology. The one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369036.png" /> of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369037.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369038.png" />, which is continuous by virtue of the linearity of the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369039.png" />, does not have, in general, a continuous inverse. If this mapping is a homeomorphism, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369040.png" /> is the direct topological sum of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369042.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369043.png" /> is said to be the direct topological complement of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369044.png" />, the latter being known as a complementable subspace. Not all subspaces in an arbitrary linear topological space, not even the finite-dimensional ones, are complementable. The following necessary and sufficient condition for complementability holds: The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369045.png" /> is topologically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369047.png" /> is an algebraic complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369048.png" />. This criterion entails the following sufficient conditions for complementability: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369049.png" /> is closed and has finite codimension; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369050.png" /> is locally convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369051.png" /> is finite-dimensional; etc.
 
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369031.png" /> be a linear topological space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369032.png" /> be the direct algebraic sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369033.png" /> of its subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369035.png" />, regarded as linear topological spaces with the induced topology. The one-to-one mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369036.png" /> of the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369037.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369038.png" />, which is continuous by virtue of the linearity of the topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369039.png" />, does not have, in general, a continuous inverse. If this mapping is a homeomorphism, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369040.png" /> is the direct topological sum of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369042.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369043.png" /> is said to be the direct topological complement of the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369044.png" />, the latter being known as a complementable subspace. Not all subspaces in an arbitrary linear topological space, not even the finite-dimensional ones, are complementable. The following necessary and sufficient condition for complementability holds: The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369045.png" /> is topologically isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369047.png" /> is an algebraic complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369048.png" />. This criterion entails the following sufficient conditions for complementability: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369049.png" /> is closed and has finite codimension; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369050.png" /> is locally convex and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369051.png" /> is finite-dimensional; etc.
  
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===Hilbert spaces===
 
A special case of topological complementation is the orthogonal complement of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369052.png" /> of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369053.png" />. This is the set
 
A special case of topological complementation is the orthogonal complement of a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369052.png" /> of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369053.png" />. This is the set
  
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which is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369055.png" />. An important fact in the theory of Hilbert spaces is that any closed subspace of a Hilbert space has an orthogonal complement, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369056.png" />.
 
which is a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369055.png" />. An important fact in the theory of Hilbert spaces is that any closed subspace of a Hilbert space has an orthogonal complement, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369056.png" />.
  
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===Vector lattices===
 
Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369057.png" /> be a conditionally order-complete vector lattice (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369059.png" />-space). The totality of elements of the form
 
Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369057.png" /> be a conditionally order-complete vector lattice (a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023690/c02369059.png" />-space). The totality of elements of the form
  
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.P. Robertson,  W.S. Robertson,  "Topological vector spaces" , Cambridge Univ. Press  (1964)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.Z. Vulikh,  "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,  "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
 +
<TR><TD valign="top">[4]</TD> <TD valign="top">  A.P. Robertson,  W.S. Robertson,  "Topological vector spaces" , Cambridge Univ. Press  (1964)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  H.H. Schaefer,  "Topological vector spaces" , Macmillan  (1966)</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">  B.Z. Vulikh,  "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====Comments====
A conditionally (order-)complete vector lattice is a [[Vector lattice|vector lattice]] that is a [[Conditionally-complete lattice|conditionally-complete lattice]].
+
A conditionally (order-)complete vector lattice is a [[vector lattice]] that is a [[conditionally-complete lattice]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,  "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.A.J. Luxemburg,  A.C. Zaanen,  "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR>
 +
</table>

Revision as of 19:42, 8 December 2014


An operation which brings a subset $M$ of a given set $X$ into correspondence with another subset $N$ so that if $M$ and $N$ are known, it is possible in some way to reproduce $X$. Depending on the structure with which $X$ is endowed, one distinguishes various definitions of complementation, as well as various methods of reconstituting $X$ from $M$ and $N$.

Sets

In the general theory of sets the complement of a subset $M$ (or complementary subset, relative complement) in a set $X$ is the subset $\complement_X M$ (or $\complement M$ if $X$ is assumed, or $X \setminus M$) consisting of all elements $x \in X$ not belonging to $M$; an important property is the duality principle (one of the De Morgan laws): $$ \complement \bigcup_{\xi} M_\xi = \bigcap_\xi \complement M_\xi $$

Linear spaces

Let $X$ have a structure of a linear space and let $M$ be a subspace of $X$. A subspace $N \subset X$ is said to be a direct algebraic complement (or algebraic complement, for short) of $M$ if any $x \in X$ can be uniquely represented as $x = y+z$, $y \in M$, $z \in N$. This is equivalent to the conditions $X = M + N$; $M \cap N = \{0\}$. Any subspace of $X$ has an algebraic complement, but this complement is not uniquely determined.

Linear topological spaces

Let be a linear topological space and let be the direct algebraic sum of its subspaces and , regarded as linear topological spaces with the induced topology. The one-to-one mapping of the Cartesian product onto , which is continuous by virtue of the linearity of the topology , does not have, in general, a continuous inverse. If this mapping is a homeomorphism, i.e. if is the direct topological sum of the spaces and , the subspace is said to be the direct topological complement of the subspace , the latter being known as a complementable subspace. Not all subspaces in an arbitrary linear topological space, not even the finite-dimensional ones, are complementable. The following necessary and sufficient condition for complementability holds: The subspace is topologically isomorphic to , where is an algebraic complement of . This criterion entails the following sufficient conditions for complementability: is closed and has finite codimension; is locally convex and is finite-dimensional; etc.

Hilbert spaces

A special case of topological complementation is the orthogonal complement of a subspace of a Hilbert space . This is the set

which is a closed subspace of . An important fact in the theory of Hilbert spaces is that any closed subspace of a Hilbert space has an orthogonal complement, .

Vector lattices

Finally, let be a conditionally order-complete vector lattice (a -space). The totality of elements of the form

which is a linear subspace of , is said to be the disjoint complement of the set . If is a linear subspace, then, in the general case, , but if is a component (also known as a band or an order-complete ideal), i.e. a linear subspace such that and imply that , and such that is closed with respect to least upper and greatest lower bounds, then (the set is a component for any ; is the smallest component containing the set ).

References

[1] N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French)
[2] N. Bourbaki, "Elements of mathematics. Topological vector spaces" , Addison-Wesley (1977) (Translated from French)
[3] G. Birkhoff, "Lattice theory" , Colloq. Publ. , 25 , Amer. Math. Soc. (1973)
[4] A.P. Robertson, W.S. Robertson, "Topological vector spaces" , Cambridge Univ. Press (1964)
[5] H.H. Schaefer, "Topological vector spaces" , Macmillan (1966)
[6] B.Z. Vulikh, "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff (1967) (Translated from Russian)


Comments

A conditionally (order-)complete vector lattice is a vector lattice that is a conditionally-complete lattice.

References

[a1] G. Köthe, "Topological vector spaces" , 1 , Springer (1969)
[a2] W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971)
How to Cite This Entry:
Complementation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complementation&oldid=35505
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article