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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[Banach spaces|06GZ956657]] over $\\R$ or $\\C$, with [[continuous duals|4A82S5FVRT]] $X'$ and $Y'$.\nLet $T \\colon X \\to Y$ be a [[bounded linear map|03TM135G0N]], and let $T' \\colon Y' \\to X'$, $T'(g) = g \\circ T$, be its [[dual operator|VCHS7PH0NF]].\nThen $T$ is a [[compact operator|M2NEP1K5AE]] if and only if $T'$ is a compact operator.\n\n## Remarks\n\nFor the forward direction, let $K$ be the closure of the image of the unit ball of $X$; it is a compact metric space.\nThe functionals in the unit ball of $Y'$, restricted to $K$, form a bounded and equicontinuous family of continuous functions on $K$.\nBy the Arzelà–Ascoli theorem, every sequence $(g_n)$ in this ball has a subsequence that converges uniformly on $K$, and then $(T' g_n)$ converges in $X'$.\nFor the converse, apply the forward direction to $T'$: the bidual $T''$ is compact, and $T$ is its restriction to $X$ under the canonical isometric embeddings $X \\subseteq X''$ and $Y \\subseteq Y''$.\nIn a Hilbert space $H$, the same theorem says that a bounded operator $T$ is compact if and only if its Hilbert space adjoint $T^*$ is compact.\n\n## Explanations\n\n### L1 human\n\nA compact operator squeezes the unit ball into a set that is almost finite-dimensional.\nEvery operator $T$ has a partner $T'$ that acts on linear functionals, the dual operator.\nThe theorem says that $T$ squeezes in this strong way exactly when its partner does.\n\n### L2 human\n\nThe theorem lets one move compactness between a space and its dual.\nIt is a key step in the Riesz–Schauder theory of compact operators: it shows that $T - \\lambda$ and $T' - \\lambda$ have kernels of the same finite dimension for $\\lambda \\ne 0$.\nIt also shows that the compact operators on a Hilbert space form a closed ideal stable under the adjoint, which is why the quotient by them is a C*-algebra.\n",
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