I am considering two metric spaces:
The Euclidean plane $\mathbb R^2$, equipped with the Euclidean distance metric $d((x, y), (x^\prime, y^\prime)) = \sqrt{(x - x^\prime)^2 + (y - y^\prime)^2}$, and
The sequence space $\ell_1$ with the $\ell_1$ metric $d((x_1, x_2, \dots), (x_1^\prime, x_2^\prime, \dots)) = \sum_i |x_i - x_i^\prime|$.
My question is whether there exists an isometric embedding from $\mathbb R^2$ to $\ell_1$.
My attempt To investigate this, I considered a simple function
$$i: (x, y) \in \mathbb R^2 \mapsto (x, y, 0, 0, \dots ) \in \ell_1$$
However, it is clear that this mapping does not preserve distances, and therefore, it is not an isometric embedding. I'm looking for alternative ways to approach this problem.
If an isometric embedding is not possible, I would like to know whether an "almost isometric" embedding (an embedding that preserves distances up to a small error) might exist between these two spaces. Any insights or references related to this question would be appreciated.