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  "body": "\n## Summary\n\nA nest is a totally ordered and complete family of closed subspaces of a Hilbert space.\nThe nest algebra of a nest is the algebra of all bounded operators that leave every subspace of the nest invariant.\nNest algebras are the infinite-dimensional analogues of the algebras of upper triangular matrices.\nThey are the main examples of non-self-adjoint operator algebras, and every nest algebra is a [[reflexive operator algebra|672HNBV803]].\n\n## Explanations\n\n### L1 human\n\nAn upper triangular $n \\times n$ matrix sends the span of the first $k$ basis vectors into itself, for each $k$.\nThese spans form a chain of subspaces, each inside the next.\nA nest algebra keeps this picture but allows any complete chain of closed subspaces of a Hilbert space, possibly infinite and with no basis behind it.\nFor example, on $L^2[0, 1]$ the subspaces of functions vanishing on $[0, t]$, for $0 \\le t \\le 1$, form a nest, and the Volterra integration operator leaves each of them invariant.\n",
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