Completeness and the Contraction Mapping Theorem
Say what a complete metric space is, and prove that a contraction on one has exactly one fixed point that iteration finds.
What a learner can do afterwards
- Give a metric space that is not complete and identify the missing limit
- Prove the contraction mapping theorem and extract the error bound iteration gives
- Apply the theorem to prove existence and uniqueness for a differential equation
1 · Read
A metric space is complete when every Cauchy sequence converges to a point inside. Gaps can fool you otherwise. Reciprocals 1, 1 over 2, 1 over 3 squeeze together yet aim at 0, missing from the half open interval that excludes it. Rationals aiming at square root of 2 miss the same way. Closed intervals keep all limits and dodge the gap.
A contraction shrinks every distance by a fixed factor k below 1. The map sending x to x plus 4 over 3 uses k equal 1 over 3. Iteration then forms a Cauchy sequence, and each step multiplies the gap by at most k. Summing the tail bounds the distance left, so halving maps turn a move of 8 into at most 4 next. The map halving then adding 1 rests at 2.
Banach fixed point theorem says a contraction on a complete space owns exactly one fixed point, and iteration from anywhere finds it. Completeness catches the orbit while shrinking forbids a rival. The same machine proves differential equations have one solution: rewrite the equation as a contraction on a complete function space and let iteration converge.
Demand k strictly below 1. A factor of exactly 1 never forces the tail to summable size. When an iteration drifts, first suspect a missing limit, then suspect a loose constant.
Completeness catches Cauchy orbits, and shrinking maps own one rest point.
More to read
2.4 Continuity
Continuity controls how maps distort distance. A contraction shrinks every distance by a fixed factor below one. Iteration then forms a Cauchy sequence. Completeness supplies the limit inside the space. Without completeness, the same iteration can aim at a missing point.
2.2 The Limit of a Function
Limits make iteration precise. Error bounds use geometric series in the contraction factor. Each step multiplies the previous gap by at most k. Summing the tail bounds distance to the fixed point. This quantitative control is what makes Banach iteration useful for equations.
5.1 Sequences
Sequences carry the convergence argument. Reciprocals in a half open interval show how Cauchyness need not imply convergence when the limit is missing. Rationals approximating an irrational show the same gap. Closed intervals avoid the gap by containing all limits.
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Where it sits
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.