I had a question that what are all the possible metrics on the space $C[0,1]$ of all continuous functions on $[0,1]$. This question arose when I was trying to prove that
$$\mathscr{F} := \left\{F(x) = \int_{0}^{x}f(t) \mathrm{d}t\ \bigg{|}\ f \in C[0,1], \|f\|_\infty \le 1 \right\} $$ is bounded and equicontinuous subset of $C[0.1].$ Now I could prove this if I know a particular metric however I don't know what is the metric in this context on $C[0,1]$. So it would be nice if I know some metrics on the space (except the usual, defined as $d(f,g) = \int_0^{x} |f(t) - g(t)|\ \mathrm{d}t$). Or is this true for any metric? A more general question comes to the mind that for a given metric space (with one metric given), how many metrics can one define on it? (I know this seems a little vague, I would grateful if someone can edit my question). Thanks in advance.