Lebl-Basic-Analysis-I · Problem 7.5.9

Metric Spaces

Mathematics · continuous-functions

Problem

Take the metric space of continuous functions

C\bigl([0, 1], \mathbb{R}\bigr)

. Let

k \colon [0, 1] \times [0, 1] \to \mathbb{R}

be a continuous function. Given

f \in C\bigl([0, 1], \mathbb{R}\bigr)

define

\varphi_f(x) \colonequals \int_0^1 k(x, y) f(y)\, dy.

a) Show that

T(f) \colonequals \varphi_f

defines a function

T \colon C\bigl([0, 1], \mathbb{R}\bigr) \to C\bigl([0, 1], \mathbb{R}\bigr)

. b) Show that

T

is continuous.