Metric spaces are:
a) Let $X = C[0,1]$ and $$d(x,y) = \sup_{0\le t \le1} t \vert x(t) - y(t)\vert$$
b) Let $$X = \{x \in C[0,1]: \sup_{0\le t \le1}t^{-2}\vert x(t)\vert < \infty\}, d(x,y) = \sup_{0\le t \le1} \vert x(t) - y(t)\vert$$
I thought that completion of the first one will have the functions that diverge at $t = 0$ with some restriction, and the second one will be the set of functions that are $0$ at $t = 0$. I could not find a way to solve it properly. Any help will be appreciated.
Answer for b): The completion is $\{x\in C[0,1]: x(0)=0\}$ with the sup norm. If $x \in C[0,1]$ with $x(0)=0$ define $x_n(t)=0$ for $0\leq t \leq \frac 1 n$, $x_n(t)=x(t)$ for $\frac 2 n \leq t \leq 1$ and let $x_n$ have a straight line graph in $[\frac 1 n, \frac 2 n]$. Then $x_n \in X$ and $x_n \to x$ in sup norm.
The completion for $X$ in part a) is $\{x \in C(0,1]: tx(t) \, \text {is bounded}\}$. [Notice that I have $C(0,1]$ instead of $C[0,1]$]. I will let you supply a proof using an idea similar to the one for part b).