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  "body": "\n## Summary\n\nThe continuous dual space of a topological vector space $E$ is the vector space $E'$ of all continuous [[linear forms|07YQ1C78G0]] on $E$.\nIt is a subspace of the algebraic [[dual space|07JBSZE0V4]], usually much smaller in infinite dimension.\nFor a normed space it is a Banach space with the dual norm.\nDuality theory studies $E$ through $E'$ and the topologies on both, such as the [[weak topology|P6KHVSTPZV]] and the [[strong dual|RG52DZ1A8Y]] topology.\n\n## Explanations\n\n### L1 human\n\nA linear form assigns a number to each vector, in a linear way.\nIn infinite dimension most linear forms are wild, so one keeps only the continuous ones.\nSmall changes of the vector should give small changes of the number.\nFor example, on continuous functions on $[0, 1]$, the map $f \\mapsto \\int_0^1 f$ and the map $f \\mapsto f(1/2)$ are continuous linear forms for uniform convergence.\n",
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