Let $C[a,b]$ denote all continuous functions $:[a,b] \rightarrow \mathbb{R}$, with the metric
$$d(f,g)=\sup |f(x) – g(x)| \text{ for } a\leq x \leq b.$$
Show that $d(f,g)$ is a metric space.
I have started with this:
Since continuos function is bounded, so i use the same proof from it: " $B[a,b]$ denote for all bounded functions $:[a,b] \rightarrow \mathbb{R}$, with
$d(f,g)=\sup |f(x) – g(x)|$ for $a\leq x \leq b$ "
to prove those (continuos function is metric space). Is it right? Or there is another way?
Best Answer
It sounds like you already know that the set of bounded functions, $B[a,b]$, is a metric space under the metric $d$. In that case, you made a very nice observation that since continuous functions are bounded, $C[a,b]$ is contained in $B[a,b]$. And any subset of a metric space is a metric space! Well done.
You can also show it directly -- nonnegativity and symmetry are almost immediate, and then you only have to show triangle inequality.