So far we have been concerned with maps from an open subset of ℝn into ℝm. Soon we shall be considering maps from a set that is a subset of ℝn into that very set, what are often called self maps of a set. For example, the map T:[0, 1]→[0, 1] given by Tx = 1 – x is a self map. A trivial example would be the identity map T given by Tx = x on any set X whatsoever. What we shall need is a property of a special kind of self maps called contractions or contraction maps of a closed subset of ℝn (Theorem 4-1.6 below). Before proceeding to the theorem, we illustrate the ideas involved.
Partial Derivative Open Subset Jacobian Matrix Closed Subset Open Ball
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