Geodesic
On a Theorem of Mitchell
Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 215-217

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In this note we apply a theorem of T. Mitchell on fixed points of contraction maps to a particular set of contraction maps of metric spaces. Also, part of Mitchell's theorem is extended to permit application, under suitable conditions, when the space involved is not compact.We employ the following terminology due to Mitchell [1]. Let S be a semigroup; m(S) the space of all bounded real functions on S with sup norm; and X a subset of m(S). For all s ∊ S, the left translation ls {right translation rs } of m(s) by s is given by (lsf)(s') = f(ss') {(rsf)(s')=f(s's)} where f ∊ m(S) and s' ∊ S. 
Bishop, E. R. On a Theorem of Mitchell. Canadian mathematical bulletin, Tome 13 (1970) no. 2, pp. 215-217. doi: 10.4153/CMB-1970-043-9
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     title = {On a {Theorem} of {Mitchell}},
     journal = {Canadian mathematical bulletin},
     pages = {215--217},
     year = {1970},
     volume = {13},
     number = {2},
     doi = {10.4153/CMB-1970-043-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-043-9/}
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