= Closed-range bound on the kernel complement
{title2=$\|Tv\|\ge b\|v\|\quad(v\perp\ker T)$}
For a <bounded linear operator> $T$ between <Hilbert spaces>, its range is closed exactly when there is $b>0$ with $\|Tv\|\ge b\|v\|$ for all $v\in(\ker T)^\perp$. The restriction to this <orthogonal complement> is a bounded bijection onto the range, so the <bounded inverse theorem> proves necessity. Conversely, the bound makes preimages of a <Cauchy sequence> of image points a <Cauchy sequence>, proving closedness by completeness.
Back to article page
