Complementation

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 $M$ of a Hilbert space $H$. This is the set $$ N^\perp = \{ x \in H : (x,y) = 0 \ \text{for all}\ y \in N \} $$ which is a closed subspace of $H$. An important fact in the theory of Hilbert spaces is that any closed subspace of a Hilbert space has an orthogonal complement, $H = N \oplus N^\perp$.

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)