= Atkinson theorem
{c}
{title2=$T\text{ Fredholm}\iff[T]\in\mathcal Q(H)^\times$}
A <bounded operator> is <Fredholm> if and only if it has a bounded two-sided inverse modulo <compact operators>. For the forward implication invert its restriction from the orthogonal kernel complement to its closed range and extend by zero. Conversely a parametrix makes the identity compact on the kernel, forcing finite dimension. A failure of a lower bound on the kernel complement would yield unit vectors $x_n$ with $Tx_n\to0$; the compact error then forces a norm-convergent subsequence, whose nonzero limit lies both in the kernel and its complement. This proves closed range; applying the other parametrix identity to the adjoint gives finite cokernel.
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