Exercise 8.2.2

Question

Let $C[0,1]$ be the collection of continuous functions on the closed interval $[0,1]$. Decide which of the following are metrics on $C[0,1]$.
\begin{enumerate}[label=(\alph*)] 
\item $d(f,g) = \sup{\left|f(x) - g(x)\right| : x \in [0,1]}$
\item $d(f,g) = \left|f(1) - g(1)\right|$
\item $d(f,g) = \int^1_0 \left|f-g\right|$
 \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
\item The first two properties are trivial. For the triangle inequality:
$$\sup |f-g| \leq \sup (\left|f-h\right| + \left|h-g\right|) \leq \sup |f-h| + \sup |h-g|$$
\item The first property fails, e.g. $f(x) = 1$, $g(x) = x$
\item Clearly $\int^1_0 |f-g| \geq \int^1_0 0 = 0$, and if $f = g$ then $\int^1_0 |f-g| = 0$. If $\int^1_0 |f-g| = 0$, let $F(x) = \int^x_0 \left|f - g\right|$. By the Fundamental Theorem of Calculus, and noting that $\left|f-g\right|$ is continuous,
$$F'(x) = |f-g| = 0$$
implying that $f = g$. This indicates the first property is met. The second property is trivially true. The third property follows from the triangle inequality on absolute values.
 \end{enumerate}

Exercise 8.2.2

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Exercise 8.2.2

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