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  "body": "\n## Summary\n\nA closed subspace of a Banach space is complemented when it has a closed [[complementary subspace|0M83V17KHX]].\nEquivalently, it is complemented when it is the range of a bounded linear projection.\nIn linear algebra every subspace has a complement, but in Banach spaces a closed complement may fail to exist.\nComplemented subspaces are the pieces into which a Banach space splits as a direct sum.\nMuch of the isomorphic theory of Banach spaces studies them.\n\n## Explanations\n\n### L1 human\n\nIn 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.\nIn an infinite-dimensional Banach space one wants the second piece to be closed as well, so that the splitting is continuous.\nIn a Hilbert space this always works, using the orthogonal complement.\nIn 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.\n",
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