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  "body": "\n## Summary\n\nA partial isometry is a bounded operator on a Hilbert space that is isometric on the orthogonal complement of its kernel.\nIt maps an initial subspace isometrically onto a final subspace and kills everything orthogonal to the initial subspace.\nPartial isometries appear in the [[polar decomposition|0J2C7BDJQA]] $T = V\\abs{T}$ of an operator and in the comparison of projections in von Neumann algebras.\n\n## Explanations\n\n### L1 human\n\nA partial isometry first throws away part of a vector and then moves the rest rigidly.\nOn $\\C^2$, the map $(x_1, x_2) \\mapsto (0, x_1)$ is an example: it forgets $x_2$ and moves $x_1$ to the second slot without changing its size.\nIsometries, unitary operators and orthogonal projections are all partial isometries.\n",
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