Can we make $\mathbb R$ and $\mathbb R^2$ "equal" as normed vector spaces? as topological spaces?

Question

We want to know if we can make $\mathbb R$ and $\mathbb R^2$ to be equal as normed vector spaces, by choosing appropriate norms? Can they be equal as topological spaces?

• Two normed vector spaces are equal if there exists a bijective linear map between them, that preserves the norm. Since $\dim \mathbb R=1$ and $\dim \mathbb R^2=2$, then there is no bijective linear map between them (rank theorem).

• For the topological space aspect, I need to show if there is an homeomorphism or not between $\mathbb R$ and $\mathbb R^2$. I struggle to start somewhere since I cannot think of any obvious homeomorphism between them

Answer 1

Firstly, a note on terminology. They are certainly not equal and can never be made equal. Unequal things can never become equal. The question is whether they are isomorphic with respect to a suitable notion of structure.

The argument you gave in bullet point 1 shows that $\mathbb R$ and $\mathbb R^2$ are not isomorphic as linear spaces. Certainly, then, they are not isomorphic as normed spaces.

For the second bullet point, your inability to even start thinking of a homeomorphism should send you looking for a proof that none exist. So, you want to find some topological invariant that one of the spaces has, but the other does not. Just like you did in the first bullet point; the dimension of a linear space is an invariant under isomorphisms. For topological spaces the notion of dimension is tricker, but it is true that these spaces have the expected dimensions as topological spaces, dimension is invariant under homeomorphisms, and so they are not homeomorphic. However, as suggested in a hint, there is a simpler way in this case. Show that the property "removing a single point results in a connected space" is invariant under homeomorphisms, one of the spaces has that property, while the other does not.