6) Contraction Mapping Theorem

Question 1

What is the Cantor Set

Answer 1

• Start with the closed interval [0, 1].
• Remove the Open Middle Third: Remove the open interval (1/3, 2/3), leaving two closed intervals: [0, 1/3] and [2/3, 1].
• Repeat Indefinitely: Repeat this process for each remaining closed interval. In the next step, remove the middle third of each of these intervals, which are (1/9, 2/9) and (7/9, 8/9), and so on

Question 2

How is the Cantor set characterized in terms of ternary expansions

Answer 2

The Cantor set K consists of all real numbers which
have a ternary expansion containing only 0s and 2s

Question 3

What are some of the properties of the Cantor Set

Answer 3

• Uncountable
• Closed
• Compact

Question 4

Describe the proof that the Cantor set is complete

Answer 4

The Cantor set, K, is a closed subset of R
A closed subset of a complete space is complete
Since R is complete the Cantor set is complete

Question 5

What is a contraction

Answer 5

Given any metric space (X, d), a self-map f : X →
X is a contraction whenever there exists a constant 0 < K < 1 such that d(f(x), f(y)) ≤ Kd(x, y) for all x, y ∈ X.

Question 6

Is a contraction continous

Answer 6

Yeah man

Question 7

What is a fixed point of a contraction

Answer 7

A fixed point of a self-map f : X → X is a point x ∈ X for which f(x) = x

Question 8

What is The Contraction Mapping Theorem

Answer 8

Let (X, d) be a complete metric space, and f : X → X a contraction; then f has a unique fixed point