= Solution
A <metric space> metric $d:X\times X\to[0,\infty)$ satisfies positivity, $d(x,y)=0$ exactly when $x=y$, symmetry, and the <triangle inequality>. A subset $U\subseteq X$ is an <open set> when every $x\in U$ has some $\varepsilon>0$ with the open ball $B_d(x,\varepsilon)\subseteq U$.
Both $\varnothing$ and $X$ are open. An arbitrary union of open sets is open because a point lies in one member supplying a ball. A finite intersection is open because the minimum of the finitely many available radii supplies a ball. Thus the metric-open sets satisfy the <topology axioms>.
For $C[0,1]$, the <Cauchy-Schwarz inequality> gives $d_1(f,g)\leq d_2(f,g)$, so the $d_2$ topology is at least as fine as the $d_1$ topology. To see strictness, define the continuous triangular spikes
$$
f_n(x)=\sqrt n\max\{1-nx,0\}.
$$
Then
$$
d_1(f_n,0)=\frac1{2\sqrt n}\longrightarrow0,
\qquad
d_2(f_n,0)^2=\int_0^{1/n}n(1-nx)^2\,dx=\frac13.
$$
Thus $f_n$ converges to zero in the <L1 norm> but not in the <L2 norm>. \b[The two metrics induce different topologies.]
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