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  "body": "\n## Summary\n\nThe quotient of a normed space $X$ by a closed subspace $M$ is the [[quotient vector space|0BR3FY5Q8M]] $X/M$.\nIts norm measures the distance of a vector to $M$.\nIt is a normed space, and it is a [[Banach space|06GZ956657]] when $X$ is.\nQuotients appear whenever one ignores a subspace.\nExamples are the open mapping theorem, the study of the kernel of an operator, and duality, where $(X/M)^*$ is identified with the annihilator of $M$.\n\n## Explanations\n\n### L1 human\n\nTo form the quotient, one declares two vectors equal when their difference lies in $M$.\nThe size of a class is the shortest length of a vector in that class, that is, the distance from the class to the origin.\nIn the plane with the usual length, take $M$ to be the horizontal axis.\nEach class is a horizontal line, and its size is its height above or below the axis.\n",
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