Is The Banach Fixed Point Theorem True?

Let (M,d)(M,d) be a complete metric space. A contraction is a function f:MMf: M \to M for which there exists some constant 0<c<10 < c < 1 such that d(f(x),f(y))<cd(x,y)d\big(f(x), f(y)\big) < c \cdot d(x,y) for all x,yMx, y \in M.

Answer the following yes-no questions:

  • If f:MMf: M \to M is a contraction, does ff have a fixed point? ((I.e., is there some xMx\in M such that f(x)=x?)f(x) = x?)

  • If f:MMf: M \to M has a fixed point, is ff a contraction?


Hint: The first question is much harder than the second. In fact, the answer is yes, and this extremely important result is known as the Banach fixed point theorem.

To prove it, choose an arbitrary x0Mx_0 \in M and set xn=T(xn1)x_n = T(x_{n-1}) for n1n\ge 1. Then, show that xnx_n converges to some xMx\in M and that xx is the desired fixed point.

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