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  "body": "\n## Definition\n\nLet $X$ be a [[Banach space|06GZ956657]] over $\\R$ or $\\C$, and let $M$ be a [[closed|0ZCK8DQYN1]] [[linear subspace|00YC1D652K]] of $X$.\nThe subspace $M$ is *complemented* in $X$ if there is a closed linear subspace $N$ of $X$ such that $M \\cap N = \\set{0}$ and $M + N = X$.\nIn other words, $M$ and $N$ are [[complementary subspaces|0M83V17KHX]].\n\n## Remarks\n\nEquivalently, $M$ is complemented if and only if there is a continuous linear map $P \\colon X \\to X$ with $P^2 = P$ and $P(X) = M$, a bounded [[projection|033DA1WE8Y]] onto $M$.\nGiven such $P$, take $N = \\ker P$; given $N$, the algebraic projection onto $M$ along $N$ has closed graph, so it is continuous by the [[closed graph theorem|GXM8BX8YM7]].\nExample: in a [[Hilbert space|HM2HHE3J51]] every closed subspace $M$ is complemented by its orthogonal complement $M^\\perp$, which is closed with $M \\cap M^\\perp = \\set{0}$ and $M + M^\\perp$ equal to the whole space.\nEvery finite-dimensional subspace of a Banach space is complemented: extend a dual basis by the [[Hahn–Banach theorem|QB5AE3VYZZ]] to build the projection.\nNon-example: $c_0$ is a closed subspace of $\\ell^\\infty$ that is not complemented; every closed subspace $N$ with $c_0 \\cap N = \\set{0}$ has $c_0 + N \\ne \\ell^\\infty$.\nA theorem of Lindenstrauss and Tzafriri says that a Banach space in which every closed subspace is complemented is isomorphic to a Hilbert space.\n",
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