---
tag: Q0TY362K0V
kind: concept
name: Complemented subspace
msc: [46B20, 46B03]
sources:
  - url: https://en.wikipedia.org/wiki/Complemented_subspace
same_as:
  wikidata: Q137454148
  wikipedia: Complemented subspace
---

## Summary

A closed subspace of a Banach space is complemented when it has a closed [[complementary subspace|0M83V17KHX]].
Equivalently, it is complemented when it is the range of a bounded linear projection.
In linear algebra every subspace has a complement, but in Banach spaces a closed complement may fail to exist.
Complemented subspaces are the pieces into which a Banach space splits as a direct sum.
Much of the isomorphic theory of Banach spaces studies them.

## Explanations

### L1 human

In the plane, a line through the origin can be paired with another line so that every vector is a unique sum of one vector from each line.
In an infinite-dimensional Banach space one wants the second piece to be closed as well, so that the splitting is continuous.
In a Hilbert space this always works, using the orthogonal complement.
In other Banach spaces it can fail: the space $c_0$ of null sequences sits inside the space $\ell^\infty$ of bounded sequences with no closed complement.