Let $\operatorname{id}$ be the identity map of $C[0, 1]$ . Pick out the true statements.

Question

Let $V_i=(C[0, 1], d_i), i = 1, 2$ be the metric spaces where

$$d_1(f, g) = \sup_{x∈[0,1]} |f(x) − g(x)|\\ d_2(f, g) =\int_{0}^{1}|f(x) − g(x)|dx \,$$

Let $\operatorname{id}$ be the identity map of $C[0, 1]$ . Pick out the true statements.

• a) $\operatorname{id} : V_1 \to V_2$ is continuous.
• b) $\operatorname{id} : V_2 \to V_1$ is continuous.
• c) $\operatorname{id} : V_1 \to V_2$ is a homeomorphism

i know that $d_1 \ge d_2$ so option a) and c) is correct

is It true ??

THanks u

Answer 1

You are correct for a)

Hint for b):

Check continuity at $f(x)=x^n$ and $g(x)=0$