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  "body": "\n## Summary\n\nA frame of a [[Hilbert space|HM2HHE3J51]] is a family of vectors that recovers every vector stably from its inner products with the family.\nUnlike a basis, a frame may be redundant.\nA tight frame is a frame whose lower and upper frame bounds are equal.\nThen reconstruction is as simple as for an [[orthonormal basis|0S9GGTR0EX]], up to a constant factor.\nTight frames are used in signal processing, coding and quantum information, where redundancy gives robustness.\n\n## Explanations\n\n### L1 human\n\nIn the plane, take three unit vectors at angles of $120^\\circ$ to each other, like the Mercedes-Benz star.\nThey are not a basis, since there are three of them in a two-dimensional space.\nStill, for every vector $f$ the sum of the squares of its three components along them is exactly $\\tfrac32 \\norm{f}^2$.\nThis is the defining property of a tight frame: the energy of $f$ is spread evenly over the frame, whatever direction $f$ has.\n",
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