# Contraction mapping in $C([0,1])$

Suppose $$T$$ is an operator on $$C([0,1])$$ defined by $$(Tu)(t) = \displaystyle\int_{0}^{t} u(x)^2\,\mathrm dx$$. Show that $$T$$ is a contraction mapping on the closed ball of radius $$\dfrac14$$ in $$C([0,1])$$.

From a different thread (Regarding integral operators being contractions) it was recommended to use the fact that $$u^2 - v^2 = (u+v)(u-v)$$ but it's getting me nowhere. Been away from functional analysis for a while and having some trouble getting back into it. Any help is appreciated.

$$|\int (u-v) (u+v)| \leq \sqrt {\int (u-v)^{2}} \sqrt {\int (u+v)^{2}}$$. For $$u$$ and $$v$$ in the given ball $$|u+v| \leq \frac 1 2$$ so $$\sqrt {\int (u+v)^{2}} \leq \frac 1 2$$. Of course $$\sqrt {\int (u-v)^{2}} \leq \|u-v\|$$.