Research notes | Open Access
James R. Holub, "A note on best approximation and invertibility of operators on uniformly convex Banach spaces", International Journal of Mathematics and Mathematical Sciences, vol. 14, Article ID 567578, 4 pages, 1991. https://doi.org/10.1155/S0161171291000832
A note on best approximation and invertibility of operators on uniformly convex Banach spaces
It is shown that if is a uniformly convex Banach space and a bounded linear operator on for which , then is invertible if and only if . From this it follows that if is invertible on then either (i) , or (ii) is the unique best approximation to from , a natural (partial) converse to the well-known sufficient condition for invertibility that .
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