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Erschienen in:
2011 | OriginalPaper | Buchkapitel
Inverse and Implicit Function Theorems
verfasst von : Satish Shirali, Harkrishan Lal Vasudeva
Erschienen in: Multivariable Analysis
Verlag: Springer London
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So far we have been concerned with maps from an open subset of ℝ
n
into ℝ
m
. Soon we shall be considering maps from a set that is a subset of ℝ
n
into that very set, what are often called
self map
Self map
s
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 ℝ
n
(Theorem 4-1.6 below). Before proceeding to the theorem, we illustrate the ideas involved.