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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.

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    $\begingroup$ what are the topologies on $A$ and $B$? Are $A$ and $B$ the collections of all real-valued functions on $\mathbb{R}$ and $\mathbb{R}^{2} $ respectively, or do you put some limitations, like continuous functions? $\endgroup$ – erz Feb 2 at 2:09
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    $\begingroup$ A couple of basic observations that might help to sharpen the question: (1) if you consider the Banach spaces of all bounded functions, 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
  • $\begingroup$ I guess the functions I'm interested in are if f(x) are all continuous or even better $C^2$ or analytic. So in Nate's answer, f being continuous would lead $\phi(f)$ to being discontinuous. So that isometric bijection would not be what I'm looking for. I'll edit the post to reflect this $\endgroup$ – Joe Feb 2 at 2:30
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As soon as your spaces $A$ and $B$ are Banach, and the only thing which you want from $\phi$ is continuity, you can use Kadets's theorem (implicitly mentioned by Nate), or its generalization by Torunczyk to get a positive result. If they are incomplete, then you get into the area of General topology, for which you can look into Kuratowski's book.

If you want to get more than continuity from $\phi$, you get into the area, whose development is described in Benyamini-Lindenstrauss Geometric Functional Analysis.

If you would like $\phi$ to be linear, there are both positive results of this type, one of the famous is Milyutin (1966), and negative, one of the famous is Kislyakov (1975).

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