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  "body": "\n## Definition\n\nLet $H$ and $K$ be [[Hilbert spaces|HM2HHE3J51]] and let $V \\colon H \\to K$ be a bounded linear operator.\nThe operator $V$ is a *partial isometry* when\n$$\\norm{Vx} = \\norm{x} \\quad \\text{for all } x \\in (\\ker V)^\\perp,$$\nwhere $\\ker V$ is the [[kernel|02YFME3RGW]] of $V$ and $(\\ker V)^\\perp$ is its [[orthogonal complement|079D68S3AW]].\nThe space $(\\ker V)^\\perp$ is the *initial space* of $V$, and the range of $V$ is its *final space*.\n\n## Remarks\n\nExample: the backward shift $V(x_0, x_1, x_2, \\ldots) = (x_1, x_2, \\ldots)$ on $\\ell^2(\\N)$ has kernel spanned by $(1, 0, 0, \\ldots)$.\nA vector orthogonal to the kernel has $x_0 = 0$, and then $\\norm{Vx}^2 = \\sum_{n \\ge 1} \\abs{x_n}^2 = \\norm{x}^2$; so $V$ is a partial isometry.\n\nNon-example: $T = 2I$ on $\\C$ has kernel $\\set{0}$, so its initial space is $\\C$, but $\\norm{T 1} = 2 \\ne 1$.\n\nEquivalent forms: $V$ is a partial isometry if and only if $V^*V$ is an orthogonal projection.\nThen $V^*V$ is the projection onto the initial space and $VV^*$ is the projection onto the final space.\nAnother equivalent condition is $VV^*V = V$.\nThe final space of a partial isometry is closed.\n",
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