I know that all $d_p$ metrics are topologically equivalent on $\mathbb{R}$. I also know it is not the case for C[0,1] (space of continuous functions). To show that I can use a sequence of functions $f_n = x^n$, which converges in (a topology induced by) $d_1$, but it doesn't in $d_\infty$.
Now I would like to show that $d_1$ and $d_2$ are not topologically equivalent. I have tried and failed to find similar counterexample. I also tried to find a set which is open in one but not the other.
Any help would be appreciated, thanks.
Btw, I am aware of similar question here: Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$. It does not however fully answer my question
Best Answer
We need to show that the norms $L_1$ and $L_2$ are not equivalent on $C[0,1]$. Let's first note that $||f||_1\le ||f||_2$ (Jensen's ineq). Now the fact that the norms are not equivalent already follows from the fact that there exist sequences Cauchy under the first norm but not under the second. But let us get a concrete example. Take $f_n = n$ on $[0,1/n^2]$ and decreasing linearly t0 $0$ on $]1/n^2, 2/n^2]$, and then $0$ from $2/n^2$ to $1$. Clearly $||f_n||_1 \to 0$ but $||f_n||_2 \ge 1$ for all $n$.