Let 
be a normed space, 
and 
be algebraically complemented subspaces of 
(i.e., 
and 
), 
be the quotient map, 
be the natural isomorphism 
, and 
be the projection of 
on 
along 
. Then the following statements are equivalent:
1. 
is a homeomorphism.
2. 
and 
are closed in 
and 
is a homeomorphism.
3. 
and 
are closed and 
is a bounded projection.
The subspaces 
and 
are called topologically complemented or simply complemented if each of the above
equivalent statements holds (Constantinescu 2001, Meise and Vogt 1997).
Every finite dimensional subspace is complemented and every algebraic complement of a finite codimension subspace is topologically complemented. In a Banach
space 
,
two closed subspace are algebraically complemented if and only if they are complemented.
There are uncomplemented closed subspaces. For example, let 
be the disk algebra, i.e., the space of all analytic functions
on 
which are continuous on the closure of 
. Then the subspace of 
consisting of the restrictions of functions of 
to 
is not complemented in 
(Hoffman 1988).
The problems related to complemented subspaces are in the heart of the theory of Banach spaces and are more than fifty years old (Johnson and Lindenstrauss 2001).
-Algebras, Vol. 1: Banach Spaces. Amsterdam,
Netherlands: North-Holland, 2001.Hoffman, K.