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**Define a function S on the set of polynomials X by S(a+bx+.....)=max{|a|,|b|,.......}. define the distance (metric) on X by d(p,q)=S(p-q)**

let f: X -> X Prove that if f(p(x))=1+1/2$\int_{0}\,^{x} p(t)\,dt$, then f is a contraction mapping with no fixed points.

2) Let the function space F = C ([0,1]; [0, 1]) be endowed with the sup metric d. For every f in F define $g(x)=sinx.|x^0.5-f(x)|$. Show that there is exactly one f in F such that f=g.

**You make take it that F is complete.**

I know what I need to do for 1) but I can't seem to simplify it. For 2) I don't know.