Basic Analysis I — Real Analysis · Problem 7.5.9

Metric Spaces — Jiri Lebl

Mathematics — Real Analysis · continuous-functions

Problem

Take the metric space of continuous functions C([0,1],R)C\bigl([0,1],\mathbb{R}\bigr). Let k:[0,1]×[0,1]→Rk : [0,1] \times [0,1] \to \mathbb{R} be a continuous function. Given f∈C([0,1],R)f \in C\bigl([0,1],\mathbb{R}\bigr) define φf(x):−∫01k(x,y)f(y) dy.\varphi_f(x) \coloneq \int_0^1 k(x,y) f(y)\, dy. a) Show that T(f):−φfT(f) \coloneq \varphi_f defines a function T:C([0,1],R)→C([0,1],R)T : C\bigl([0,1],\mathbb{R}\bigr) \to C\bigl([0,1],\mathbb{R}\bigr). b) Show that TT is continuous.
Intuitive explanation

This problem asks you to verify that an integral operator—a machine that transforms one continuous function into another by "blending" it with a fixed kernel—is itself a well-behaved, continuous map. The key insight is that uniform continuity of kk on the compact square [0,1]2[0,1]^2 lets you control how small changes in ff or xx ripple through the integral, ensuring both that φf\varphi_f lands back in C[0,1]C[0,1] and that TT respects the uniform metric on function space.

Intuitive explanation

This problem asks you to verify that an integral operator—a machine that transforms one continuous function into another by "blending" it with a fixed kernel—is itself a well-behaved, continuous map. The key insight is that uniform continuity of kk on the compact square [0,1]2[0,1]^2 lets you control how small changes in ff or xx ripple through the integral, ensuring both that φf\varphi_f lands back in C[0,1]C[0,1] and that TT respects the uniform metric on function space.

Related theory

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