2010-2.md

course	course_year	question_number	tags	title	year
Analysis II	IB	2	IB 2010 Analysis II	Paper 4, Section I, G	2010

Let $S$ denote the set of continuous real-valued functions on the interval $[0,1]$. For $f, g \in S$, set

$$d_{1}(f, g)=\sup {|f(x)-g(x)|: x \in[0,1]} \quad \text { and } \quad d_{2}(f, g)=\int_{0}^{1}|f(x)-g(x)| d x$$

Show that both $d_{1}$ and $d_{2}$ define metrics on $S$. Does the identity map on $S$ define a continuous map of metric spaces $\left(S, d_{1}\right) \rightarrow\left(S, d_{2}\right) ?$ Does the identity map define a continuous map of metric spaces $\left(S, d_{2}\right) \rightarrow\left(S, d_{1}\right)$ ?