Range conditions on a linear operator

Question

While reading though some engineering literature, I came across some logic that I found a bit strange. Mathematically, the statement might look something like this:

I have a linear operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$, that is a mapping which takes functions of three variables to functions of four variables. Then, "because the range function depends on 4 variables while the domain function depends on only three", there must be redundancy in the operator $A$, that is the range of $A$ is a proper subset of $L^2(\Bbb{R}^4)$, characterized by some range conditions.

Is such a statement always true? For the specific example I am reading about (the X-ray transform), it is definitely true - in fact, the range of the operator is characterized by a certain PDE - but I can't image such a thing is true in general.

For instance, I can cook up an operator $A:L^2(\Bbb{R}^3)\rightarrow L^2(\Bbb{R}^4)$ such that the range of $A$ is dense in $L^2(\Bbb{R}^4)$: simply choose orthonormal bases $(e_j)$ and $(f_j)$ for both, then map $e_j$ to $f_j$.

Any thoughts?

Answer 1

I think it always has to be true. It is like mapping points in a flat plane to a 3D space. You get a volume 0 manifold in the 3D space if there is a 1 to 1 mapping.

Answer 2

It's not true. There is only one separable Hilbert space, up to isomorphism. Since $L^2(\mathbb{R}^3)$ and $L^2(\mathbb{R}^4)$ are both separable, they are isomorphic as Hilbert spaces. That is, there is an isometric linear mapping $A$ of $L^2(\mathbb{R}^3)$ onto $L^2(\mathbb{R}^4)$.