Complemented subspace

If X is a vector space and M and N are vector subspaces of X then X is the algebraic direct sum of M and N or the direct sum of M and N in the category of vector spaces if any of the following equivalent conditions is satisfied:

1. the canonical map M × N → X defined by (m, n) ↦ m + n is a vector space isomorphism;[1]
2. the canonical map M × N → X is bijective;
3. M ∩ N = { 0 } and M + N = X;

in this case N is called an algebraic complementary or algebraic supplementary to M in X.

If X is the algebraic direct sum of M and N and if S : M × N → X is the canonical map defined by (m, n) ↦ m + n, then the inverse of the canonical map S can be written as ${\displaystyle S^{-1}:=\left(P_{M},P_{N}\right):X\to M\times N}$ where P_M : X → M and P_N : X → N are called the canonical projections of X onto M and N, respectively.[1]

If X is the algebraic direct sum of M and N and if X is also a TVS then the canonical map S : M × N → X defined by (m, n) ↦ m + n is necessarily continuous (since addition is continuous) but its inverse ${\displaystyle S^{-1}:X\to M\times N}$ may fail to be continuous; when it is continuous then X is the direct sum of M and N in the category of TVSs. If f is a discontinuous linear functional on a locally convex Hausdorff TVS X, n is any element of X such that f(n) ≠ 0, N := span { n }, and M := ker f, then X is the algebraic direct sum of M and N but not the direct sum of M and N in the category of TVSs.

If X is a topological vector space (TVS) and if M and N are vector subspaces of X, then X is the topological direct sum of M and N or the direct sum of M and N in the category of TVSs if any of the following equivalent conditions is satisfied:

1. the canonical map S : M × N → X defined by (m, n) ↦ m + n is a TVS-isomorphism;
2. the canonical map S : M × N → X is a bijective open map;
3. X is the algebraic direct sum of M and N (i.e. the canonical map S : M × N → X is bijective) and the inverse of the canonical map, ${\displaystyle S^{-1}:X\to M\times N}$, is continuous;
4. X is the algebraic direct sum of M and N and the canonical projections P_M : X → M and P_N : X → N are continuous (where ${\displaystyle S^{-1}:=\left(P_{M},P_{N}\right)}$);
5. X is the algebraic direct sum of M and N and at least one of the two canonical projections P_M : X → M and P_N : X → N is continuous;
6. X is the algebraic direct sum of M and N and the restriction to N of the canonical projection Q : X → X/M (defined by Q(x) := x + M) is a TVS-isomorphism of N onto X/M;[2]
7. The canonical vector space isomorphism p : N → X/M is bicontinuous (meaning p and its inverse are continuous);
   • Note that this implies that any topological complement of M in X is TVS-isomorphic to X/M.

in this case N is called a (topological) complement, (topological) complementary, or (topological) supplementary to M in X and we say that M (as well as N) is a (topologically) complemented subspace of X.

Let X be a topological vector space and let M be a vector subspace of X. We say that M is a complemented subspace of X if any of the following equivalent conditions hold:

1. there exists a vector subspace N of X such that X is the direct sum of M and N in the category of TVSs;
2. there exists a continuous linear map P : X → X with range M such that P ∘ P = P;
3. there exists a continuous linear projection P : X → X with range M such that X is the algebraic sum of M and the null space of P.
4. for every TVS Y, the restriction map ${\displaystyle R:L(X;Y)\to L(M;Y)}$ is surjective, where R is defined by sending a continuous linear operator u : X → Y to the restriction ${\displaystyle R(u):=u{\big \vert }_{M}:M\to Y}$.[2]

It can be shown that if f : X → Y is a continuous linear bijection between two TVSs, then the following conditions are equivalent:

1. the kernel of f has a topological complement;
2. there exists a continuous linear map g : Y → X such that f ∘ g = Id_Y, where Id_Y : Y → Y is the identity map.[2]

For the special case of a Banach space, the following definition is equivalent to the one given above for TVSs.

Let ${\displaystyle X}$ be a Banach space and ${\displaystyle Y}$ a closed subspace of ${\displaystyle X}$. Then ${\displaystyle Y}$ is called a complemented subspace of ${\displaystyle X}$ whenever there exists another closed subspace ${\displaystyle Z}$ of ${\displaystyle X}$ such that ${\displaystyle X}$ is isomorphic to the direct sum ${\displaystyle Y\oplus Z}$; in this case ${\displaystyle Z}$ is also a complemented subspace, and ${\displaystyle Y}$ and ${\displaystyle Z}$ are called complements of each other (in ${\displaystyle X}$).