Geodesic
Approximating Fredholm operators on a nonseparable Hilbert space
Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 167-178

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This paper obtains a simple formula for the distance from a given operator to the set of invertible operators without requiring the underlying space to be separable. That formula is used to compute the distance to the Fredholm operators with a given index. These results require the further study of the concepts of essential nullity and essential deficiency, which permitted us to characterize the closure of the invertible operators. We also introduce a parameter called the modulus of Fredholmness. 
Bouldin, Richard. Approximating Fredholm operators on a nonseparable Hilbert space. Glasgow mathematical journal, Tome 35 (1993) no. 2, pp. 167-178. doi: 10.1017/S0017089500009721
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