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Picard Operators and Picard Spaces
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A contraction in a complete metric space is a Picard operator but a contractive map in a complete metric space need not even have a fixed point. We introduce the notion of a Picard space. It is a metric space in which every contractive map having a fixed point is a Picard operator. We then find an equivalent definition of a Picard space which allows us to show that every Heine-Borel space is Picard. Finally, we describe the Venn diagram for the classes of complete, Picard and Heine-Borel spaces.
Title: Picard Operators and Picard Spaces
Description:
A contraction in a complete metric space is a Picard operator but a contractive map in a complete metric space need not even have a fixed point.
We introduce the notion of a Picard space.
It is a metric space in which every contractive map having a fixed point is a Picard operator.
We then find an equivalent definition of a Picard space which allows us to show that every Heine-Borel space is Picard.
Finally, we describe the Venn diagram for the classes of complete, Picard and Heine-Borel spaces.
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