Prove that $(\{x\in \mathbb{R}^n\,: ||x||_1=1\}, d_2)$ and $(\{y\in \mathbb{R}^n\,: ||y||_2=1\}, d_2)$ are homeomorphic

Question

It seems like it would be very simple, but I’m having trouble proving the following:

Let $X =\{x\in \mathbb{R}^n\,: ||x||_1\}$ and $Y=\{y\in \mathbb{R}^n\,: ||y||_2=1\}$. Prove that $(X,||.||_2)$ and $(Y,||.||_2)$ are homeomorphic to each other.

I’m trying to find a homeomorphism in spaces using the same metric $||.||_2$ because I was trying to demonstrate that $X$ was compact in Euclidean space. (I know there are easier ways to answer this but it just came up when I was trying to prove it by showing that: “X is totally bounded and complete” or that “every sequence in X has a convergent subsequence”).

Basically, I’m having trouble finding a function from one to the other (I think I can show that it’s a homeomorphism after that though). Could anyone come up with one?

More generally, do you have any suggestions on finding functions between metric spaces before checking if they are homeomorphisms? Once I am given a function mapping one metric space to another, I can generally check whether it fulfills the necessary conditions to be a homeomorphism, but I often seem to have trouble finding the function itself.

Related to the question above, how would one prove that $(\mathbb{R}^n, ||.||_1)$ and $(\mathbb{R}^n, ||.||_2)$ are homeomorphic?

Edit: I originally proved it (I think) using sequential compactness (See below), but the question of whether there was a homeomorphism from X to Y came up while I was thinking of alternative ways to do so.

Carothers 8.8: Prove that the set $\{x\in \mathbb{R}^n\,:\, ||x||_1 = 1\}$ is compact in $\mathbb{R}^n$ under the Euclidean norm.

Answer 1

Let $f: X \rightarrow Y$ by $$f(x_1, \cdots, x_n) = \left(\frac{x_1}{\sqrt{x_1^2 + \cdots + x_n^2}}, \cdots, \frac{x_n}{\sqrt{x_1^2 + \cdots + x_n^2}}\right)$$ which is a homeomorphism.

Answer 2

The only function $X \to Y$ I can think of is $(x_1, x_2, ..., x_n) \mapsto (x_2, x_1, ..., x_n)$. If you can only think of one, it is the correct one most of the time. It is clear that it is self-inverse, so you only need to show it is continuous. Hint: projections and inclusions are continuous. Actually $X$ is not compact, consider the cover $(\{1\} \times ]-n, n[ \times ... \times ]-n, n[)_{n \in \mathbb{N}}$.

Edit: didn't see your second question. You would prove that a basis element in $(\mathbb{R}^n, ||.||_1)$ is open in $(\mathbb{R}^n, ||.||_2)$ and vice versa. You would need a description of the metrics to do this though.