Conceptualizing Contraction over $C([a,b])$

Question

Let $h\in C([a,b]), \delta\in C([a,b]\times[a,b])$ and $$L_k := \sup_{x\in[a,b]}\int_a^b|\delta(x,y)|dy < 1$$ Show that $T$ defines a contraction with konstant $L_k$ where $$T:C([a,b])\rightarrow C([a,b]),\quad f\mapsto h + \int_a^b \delta(\cdot, y)f(y)dy$$ I am finding it hard to conceptualize the problem, as the only examples of contractions I have seen are $\mathbb{R}\rightarrow \mathbb{R}$. How can one start a proof for this problem?

Answer 1

I suppose you are taking the sup norm on $C[a,b]$. For any $f,g \in C[a,b]$ we have $Tf(x)-Tg(x)=\int_a^{b} \delta (x,y)[f(y)-g(y)]dy$ so $|Tf(x)-Tg(x)| \leq L_k \sup \{|f(y)-g(y)|: a \leq y \leq b\}$. Since this is true for all $x$ we get $\|Tf-Tg\| \leq L_k\|f-g\|$. Since $L_k <1$ it follows that $T$ is a contraction.