Abstract
We observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space ell _2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.
Highlights
Functional equation, Orthogonality equation, Semi-inner product, Uniformly smooth Banach space. It is an easy consequence of the polarisation identity that unitary maps between Hilbert spaces, that is, maps preserving the inner product, are automatically linear
Since inner-product spaces are characterised by the inextricable connection between the norm and the inner product, the aforementioned fact does not have a canonical interpretation in the non-Hilbertian setting
Natural approaches to extending Uhlhorn’s version of Wigner’s theorem on symmetry transformations [5] are available in the Banach-space setting, for example, in terms of Birkhoff–James orthogonality [1] or semi-inner products [3]
Summary
Functional equation, Orthogonality equation, Semi-inner product, Uniformly smooth Banach space. It is an easy consequence of the polarisation identity that unitary maps between Hilbert spaces, that is, maps preserving the inner product, are automatically linear. We observed in [6, Theorem 7] that if X is a non-Hilbertian finite-dimensional space with dim X 3 that is smooth, there exists a space V of dimension dim X −1 and a non-linear map f : V → X that preserves semi-inner products.
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