Complemented subspace

In the branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a> called <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a complemented subspace of a <a href="/facts/Topological_vector_space/XRex1ChN">topological vector space</a> 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 is a <a href="/facts/Vector_subspace/2wtKTgHL">vector subspace</a> 
 
 
 
 M
 
 
 {\displaystyle M}
 
 for which there exists some other vector subspace 
 
 
 
 N
 
 
 {\displaystyle N}
 
 of 
 
 
 
 X
 ,
 
 
 {\displaystyle X,}
 
 called its (topological) complement in 
 
 
 
 X
 
 
 {\displaystyle X}
 
, such that 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the <a href="/facts/Direct_sum_of_topological_vector_spaces/x3ljVsP6">direct sum 
 
 
 
 M
 ⊕
 N
 
 
 {\displaystyle M\oplus N}
 
 in the category of topological vector spaces</a>. Formally, topological direct sums strengthen the <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">algebraic direct sum</a> by requiring certain maps be continuous; the result retains many nice properties from the operation of direct sum in finite-dimensional vector spaces. 
Every finite-dimensional subspace of a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> is complemented, but other subspaces may not. In general, classifying all complemented subspaces is a difficult problem, which has been solved only for <a href="/facts/List_of_Banach_spaces/6dyURAo0">some well-known Banach spaces</a>. 
The concept of a complemented subspace is analogous to, but distinct from, that of a <a href="/facts/Complement_(set_theory)/yWUePIYY">set complement</a>. The set-theoretic complement of a vector subspace is never a complementary subspace.

Complemented subspace open-in-new

Complemented subspace