It is well-known that any bijection $\mathbb{R} \rightarrow \mathbb{R}^2$ cannot be continuous. But suppose we have the two spaces $A = \{f(x):\mathbb{R^2}\rightarrow \mathbb{R} \}$ and $B = \{f(x):\mathbb{R}\rightarrow \mathbb{R} \}$ and and a map between them $A \xrightarrow{\phi} B$. (perhaps adding regularity conditions on the functions in $A,B$ so they're Hilbert or Banach spaces and continuity can be defined for $\phi$). The question is, can $\phi$ be both continuous and bijective?

This question is inspired by trying to invert the Radon transform for tensor fields, as in https://arxiv.org/pdf/1311.6167.pdf, but can be formulated outside of this context.

EDIT: The conditions I want on $A,B$ should be function spaces of continuously differentiable functions, or even stronger like analytic functions, equipped with some $L^p$ norm. These regularity conditions are to model something like the Radon or tensor transform in the sense ensuring that small perturbations to the data lead to small perturbations in the reconstructions.

allreal-valued functions on $\mathbb{R}$ and $\mathbb{R}^{2} $ respectively, or do you put some limitations, like continuous functions? $\endgroup$ – erz Feb 2 at 2:09boundedfunctions, then there's a trivial isometric bijection: let $\tau$ be your favorite bijection between $\mathbb{R}$ and $\mathbb{R}^2$, and set $\phi(f) = f \circ \tau$. (2) If you look at spaces like all continuous functions vanishing at infinity, then they are homeomorphic because all separable Banach spaces are homeomorphic. $\endgroup$ – Nate Eldredge Feb 2 at 2:19