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Unformatted text preview: ic satisfies: iff
(triangle inequality). Example. A normed linear space is a metric space with
same thing as a vector space.) ■
A sequence converges to if as . (A linear space is the . If , then convergence corresponds to uniform convergence of a sequence of functions.
is Cauchy if,
. From Analysis, every convergent sequence is Cauchy. However, not every Cauchy sequence
Example. The set of numbers with . The sequence is Cauchy but does not converge (in the sense that its limit does not belong to ). ■
A metric space is complete if every Cauchy sequence converges to an element of . A complete
normed linear space is a Banach space. Contraction Maps
Meiss uses map as a synonym for operator.
Let be a map on a complete metric space
for all . is a contraction if there is a constant
. Remark. The following little exercise in geometric series is used in the following theorem. Let
Theorem 3.4 (Contraction Mapping). Let
space . Then has a unique fixed point be a contraction on a complete metric
. Proof. As in the text. Plan to give it.
Example (source: http://www.math.uconn.edu/~kconrad/blurbs/analysis/contraction.pdf )
Consider the complete metric space
theorem applied to
leads to for and some between with . The mean value and . We then have
, is a contraction. Starting from any
, repeated applications
of gives a sequence that converges to a unique fixed point. Test this by entering on your
calculator and repeatedly pressing the cosine key. The unique fixed point
■ Lipschitz Functions
Let . A function is Lipschitz if for all
there is a constant such that
. The smallest such constant is the Lipschitz constant for in . Lemma 3.5. A Lipschitz function is uniformly continuous.
Proof. is uniformly continuous means
. But Lipschitz implies
The following example shows that such that
. Therefore is uniformly need not be differentiable in order to be Lipschitz. 6
. sketch Define piecewise by Example. . For . The same statement holds if
constant =1. ■ and , A function
Lipschitz on . is locally Lipschitz if for all Example. is not locally Lipschitz at sufficiently small that are both less than or equal to
. is Lipschitz with there is a neighborhood . For any there is is with .■ , implying Lemma 3.6 Let be a
function on a compact, convex set . Then
condition on with Lipschitz constant
and . Since such that satisfies a Lipschitz let
. parameterizes the straight line between
is convex, this line is within . sketch A, x, y, line By the Fundamental Theorem of Calculus and the chain rule: Since is compact and is continuous, has a maximum value on . Thus .□
Corollary 3.7 If is on an open set , then it is locally Lipschitz on . Proof. For each
there is a closed bal...
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- Fall '14