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Differential Equations : The Contraction Mapping Theorem

1). Define T : C[0,1] --> C[,1] by (Tx)(t) = 1 + integral from 0 to 1 x(s)ds. Is T a contraction? ( Please justify every step and claim, I want a proof not a yes or no only).
P. S. I believe C[0,1] is the set of all the continuous functions on [0,1].

2). Consider the operator in C[0,1],
Ty(t) = integral from 0 to t (t-s)*y(s)ds. Show that T is a contraction. ( Also here I want a detailed proof).

Solution Summary

Contractions are investigated. The solution is detailed and well presented.