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  "body": "\n## Summary\n\nThe compact operators on a Hilbert space $H$ form a set $\\mathcal{K}(H)$.\nIt is a norm-closed two-sided ideal of the algebra $\\mathcal{B}(H)$ of bounded operators, closed under taking adjoints.\nIt is the norm closure of the [[finite-rank operators|Z8FABSYQTS]], and it is a C*-algebra in its own right.\nFor a separable infinite-dimensional $H$, it is the only closed two-sided ideal of $\\mathcal{B}(H)$ other than $\\set{0}$ and $\\mathcal{B}(H)$, a theorem of Calkin.\nThe quotient $\\mathcal{B}(H)/\\mathcal{K}(H)$ is the [[Calkin algebra|6FE7JZ2VF1]], the setting of Fredholm theory.\n\n## Explanations\n\n### L1 human\n\nCompact operators behave like small perturbations: adding or composing with a bounded operator keeps them compact.\nThat is what it means to be a two-sided ideal: a product $AK$ or $KA$ of a bounded operator and a compact one is again compact.\nWorking modulo this ideal ignores compact perturbations, which is the right point of view for questions such as whether an operator is invertible up to a finite-dimensional error.\n",
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