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  "body": "\n## Summary\n\nThe dual operator of a bounded operator $T \\colon X \\to Y$ between normed spaces is the operator $T' \\colon Y' \\to X'$ between the continuous dual spaces given by $T'(f) = f \\circ T$.\nIt is the analytic form of the [[dual map|0AZ5ARP287]] of linear algebra, also called the transpose or Banach space adjoint.\nMany properties of $T$, such as having closed or dense range, can be read off from $T'$.\n\n## Explanations\n\n### L1 human\n\nA continuous linear functional measures vectors, giving one number for each vector.\nGiven an operator $T$ from $X$ to $Y$, any measurement $f$ on $Y$ gives a measurement on $X$: first apply $T$, then measure with $f$.\nThat is the dual operator, and it goes in the opposite direction.\nFor a matrix acting on column vectors, the dual operator is given by the transposed matrix.\n",
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