A unitary change of variables preserves expectation values because the position operator transforms with the state, so the recent anti-Hilbert-space argument fails.
and Todorov, I.T.: Introduction to Axiomatic Quantum Field Theory
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Are Hilbert Spaces Unphysical? Hardly, My Dear!
A unitary change of variables preserves expectation values because the position operator transforms with the state, so the recent anti-Hilbert-space argument fails.