DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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A complemented closed subspace of a normed space

Definition

Let $X$ be a normed space and let $M \leq X$ be a closed linear subspace. We say that $M$ is complemented when there is a closed linear subspace $N \leq X$ such that every $x \in X$ admits a unique decomposition

$x = m + n (m \in M, n \in N),$

and the coordinate maps

$P_{M} (x) : = m, P_{N} (x) : = n$

are bounded linear operators on $X$ .

Remarks

The decomposition is written $X = M \oplus N$ .
The theorem below shows that this is equivalent to $M$ being the range of a bounded projection.

Depends on

Used by

Closed finite-codimensional subspaces are complemented Corollary
Finite-dimensional subspaces are complemented Corollary
A closed subspace of ell-infinity that is not complemented Counterexample
An algebraic complement need not be a topological complement Counterexample
Smooth Banach vector bundle and section Definition
Split Banach submanifold Definition
A projection with finite-dimensional kernel is Fredholm Example
A regular level set in a Banach space Example
Fredholm splitting and parametrix Lemma
Local finite-dimensional reduction for a Fredholm map Lemma
Surjectivity alone does not imply a complemented kernel Remark
A closed subspace is complemented exactly when it is the range of a bounded projection Theorem
A split surjective derivative parametrises its level set Theorem
A transverse Banach bundle section has a split zero submanifold Theorem
c₀ is not complemented in ell-infinity Theorem
Regular value theorem for Banach manifolds Theorem
Under Dependent Choice, a surjective bounded operator between Banach spaces has a bounded right inverse exactly when its kernel is complemented Theorem

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)
Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations (standard reference, not scraped)