| course |
course_year |
question_number |
tags |
title |
year |
Analysis and Topology |
IB |
0 |
IB |
2021 |
Analysis and Topology |
|
Paper 2, Section I, $2 F$ |
2021 |
Let $K:[0,1] \times[0,1] \rightarrow \mathbb{R}$ be a continuous function and let $C([0,1])$ denote the set of continuous real-valued functions on $[0,1]$. Given $f \in C([0,1])$, define the function $T f$ by the expression
$$T f(x)=\int_{0}^{1} K(x, y) f(y) d y$$
(a) Prove that $T$ is a continuous map $C([0,1]) \rightarrow C([0,1])$ with the uniform metric on $C([0,1])$.
(b) Let $d_{1}$ be the metric on $C([0,1])$ given by
$$d_{1}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$$
Is $T$ continuous with respect to $d_{1} ?$