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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[normed spaces|03J5RAKB30]], and let $D(T)$ be a linear subspace of $X$.\nAn *operator* from $X$ to $Y$ is a [[linear map|0NRYHJPQG2]] $T \\colon D(T) \\to Y$.\n*Operator theory* is the branch of mathematics that studies operators.\nIts main objects are the [[continuous linear maps|03TM135G0N]] between [[Banach spaces|06GZ956657]] and the operators on a [[Hilbert space|HM2HHE3J51]].\nIts central questions concern invertibility, the spectrum, adjoints, invariant subspaces and the classification of operators.\n\n## Remarks\n\nExample: the right shift $S(x_1, x_2, \\ldots) = (0, x_1, x_2, \\ldots)$ on the space $\\ell^2$ of square-summable sequences is an operator.\nIt preserves norms, so it is bounded with norm $1$.\nIt is injective but not surjective, since every vector in its image has first coordinate $0$.\n\nNon-example: the map $x \\mapsto \\norm{x}^2 x$ on $\\ell^2$ is not an operator in this sense, since it is not linear.\nSuch maps belong to nonlinear functional analysis.\n\nOn a finite-dimensional space, operator theory reduces to the linear algebra of matrices.\n\n## Explanations\n\n### L1 human\n\nOperator theory is linear algebra for spaces of infinite dimension.\nIt studies maps that respect sums and scalar multiples, and that act on sequences or functions instead of finite lists of numbers.\n",
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