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  "body": "\n## Summary\n\nLeast squares approximation replaces a function by the element of a given finite-dimensional family that is closest in a mean square sense.\nIt is the [[best approximation|7N4988HN3P]] in an [[inner product space|0QVNCJXRY3]], so it is the [[orthogonal projection|0B8KZRT28M]] onto the family.\nUnlike the uniform best approximation, it is found by solving a linear system, and an orthogonal basis makes it explicit.\nIt is the function version of the [[least squares|0BYTQ4X1HC]] method for linear systems.\n\n## Explanations\n\n### L1 human\n\nMeasure the error of an approximation by squaring the gap at each point and adding up, or integrating.\nThe least squares approximation is the choice with the smallest total squared error.\nLarge gaps count a lot, but a single bad point does not decide everything, as it does for the largest-error measure.\nFor example, the best constant approximation of $x^2$ on $[-1, 1]$ in this sense is its mean value $1/3$.\n",
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