Positive linear maps that converge on a small test set of continuous functions converge on the whole space, even when the limit is a nonlinear isometry.
Altomare, Korovkin-type theorems and approximation by positive line ar operators, Surv
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Korovkin-type results on convergence of sequences of positive linear maps on function spaces
Positive linear maps that converge on a small test set of continuous functions converge on the whole space, even when the limit is a nonlinear isometry.