When are maps preserving semi-inner products linear?

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 ℓ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _2$$\end{document} 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.


Introduction
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. Nonetheless, 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]. In the present paper we focus on the latter approach.
Lumer [4] and Giles [2] proved that reminiscences of inner products are available in arbitrary normed spaces as for every normed space X one may find a pairing [ · | · ] thereon (a semi-inner product) that assumes scalar values, is linear in the first variable, anti-homogeneous in the second variable, and the following form of the Cauchy-Schwarz inequality holds: [x|y] x · y (x, y ∈ X) P. Wójcik AEM with [x|x] = x 2 . In particular, for each w ∈ X, [ · |w] ∈ X * . Semi-inner products are, in general, non-unique: a normed space X has a unique semi-inner product if and only if it is smooth, that is every nonzero vector x ∈ X admits a unique norming functional, that is, a norm-one functional ϕ x ∈ X * such that ϕ x , x = x . In the case where X is smooth, we have We observed in [6,Theorem 7] that if X is a non-Hilbertian finite-dimensional space with dim X 3 that is smooth, then there exists a space V of dimension dim X −1 and a non-linear map f : V → X that preserves semi-inner products. The map f may even be discontinuous.
The first result demonstrates that for X and Y having equal finite dimensions, without any additional hypotheses, a semi-inner product preserving function between X and Y must be a linear isometry.

Theorem 1. Let X and Y be normed spaces with fixed semi-inner products
If either (a) X and Y have the same finite dimension, or (b) X has a Schauder basis (e i ) and (f (e i )) is a Schauder basis of Y , then f is a linear isometry.
The proof will be presented in the subsequent section. We highlight Theorem 1 as clause (a) appears to be optimal in the case where no further assumptions on f are imposed in the light of the following blatant counterexample of an analogous statement in infinite dimensions.

Theorem 2. There exists a uniformly smooth renorming X of the Hilbert space
Since X is smooth, the choice of the semi-inner product is unambiguous. Regrettably, Theorem 2 refutes a side result from a recent paper by Ilišević and Turnšek [3, Proposition 2.4(ii)], where it was claimed that if X is a smooth Banach space and f : X → X is a (possibly non-surjective) map satisfying (3), then f is necessarily a linear isometry. Their proof contains a flaw as explained in Remark 6. However, the main results in [3] are dealing with surjective functions between smooth normed spaces which satisfy the Wigner equation. It can be easily verified (see [3, Proposition 2.4(i)]) that, even in arbitrary normed Our notation and terminology are standard. We consider normed spaces over the field K of real or complex numbers. A normed space X is strictly convex if the unit sphere of X does not contain non-trivial line segments. We denote by ·, · the duality pairing between a normed space X and its dual X * . When X is an inner-product space, we denote by ·|· the underlying inner product. A normed space is uniformly smooth, if for every ε > 0 there exists δ > 0 such that if x, y ∈ X are vectors such that x = 1 and y δ then x + y + x − y 2 + ε y . Uniformly smooth spaces are, in particular, smooth. For the sake of completeness, we record the following simple property of smooth spaces.

Proofs of Theorems 1 and 2
We start by proving Theorem 1. For the sake of brevity, we shall use the symbol [ · | · ] for (fixed) semi-inner products both in X and Y , hoping it will not lead to unnecessary confusion. In order to prove the theorem, it suffices to show that f is linear.
Proof of Theorem 1. We will prove clause (b) first. Suppose that (e i ) is a Schauder basis of X and (f (e i )) is a Schauder basis of Y . The proof of the case where X is finite-dimensional (so that a Schauder basis is just an ordinary algebraic basis) is mutatis mutandis the same, so we will keep writing infinite series bearing in mind that the proof works equally well for the finite-dimensional case with ∞ replaced by dim X).
For every unit vector u ∈ X, we have 1 = u = f (u) . Thus, for every m we have Using the linearity of semi-inner products in the first variable, it follows that Combining the above inequality with (2) yields Consequently, which means that [x − x m |u] ε m . Since [ · |w] ∈ X * : w = 1, w ∈ X is a 1-norming subset in the dual ball of X * , we can conclude that we have In particular, f is linear, hence also isometric because it preserves the semi-inner products. Now, in order to prove clause (a), it is enough to show that f maps linearly independent sets to linearly independent sets.
Let n = dim X. Fix a basis {b 1 , . . . , b n } for X. We claim that the set When are maps preserving semi-inner products linear? 673 Hence n k=1 α k b k = 0. Since the vectors b 1 , . . . , b n are linearly independent, we have α 1 = . . . = α n = 0. This means that the vectors f (b 1 ), . . . , f(b n ) are linearly independent too. Consequently, {f (b 1 ), . . . , f(b n )} is a basis for Y . Thus we may apply (b), and the proof is complete.
Before we prove Theorem 2, we need to introduce the main building block that we shall use to construct the sought renorming of 2 .
Let (Z, · o ) be a two-dimensional normed space that is smooth but not strictly convex. Then there are distinct vectors u, w ∈ Z such that the line segment joining u and w lies in the unit sphere S Z of Z. Without loss of generality, we may assume that Z = K 2 as a vector space and u = (−c, 1), w = (c, 1) for some real number 0 < c < 1. Thus (0, 1) ∈ S Z . Moreover, without loss of generality we may assume that (1, 0) ∈ S Z .
Thus (η, 1) ∈ S Z , i.e., (η, 1) o = 1. Since (0, 1) ∈ S Z , (0, 1) o = 1. Therefore (ηx 1 , Proof of Theorem 2. We shall consider the space X = K ⊕ 2 2 (Z), the 2 -sum of infinitely many copies of Z and the one-dimensional space. The norm in X is thus given by The space X is uniformly smooth because Z is smooth (uniformly smooth as it is finite-dimensional) and uniform smoothness passes to 2 -sums of infinitely many copies of a uniformly smooth space [7,Corollary 4.9]. Since Z is isomorphic to the two-dimensional Hilbert space, X is isomorphic to 2 ( 2 2 ), which is isometric to 2 .
Combining (1) with (6) we may rearrange (6) as Moreover, putting y in place of x in (6) we get We are now ready to construct the sought non-linear map that preserves semi-inner products.
We claim that for all x, y ∈ X we have [f (x)|f (y)] = [x|y]. For this, fix x, y ∈ X and consider the associated maps h η(y) , h η(x) : X → X. Applying again Lemma 4 to (4) and (5), we conclude that It follows from Lemma 3 that ϕ h η(y) (y) = ϕ h η(x) (y) , i.e., Consequently, This shows that f : X → X is indeed a non-linear map preserving semi-inner products.
Vol. 96 (2022) When are maps preserving semi-inner products linear? 675 Remark 5. In the above construction one may consider the p -sums for p ∈ (1, ∞) instead of the 2 -sum. This will lead to a renorming of p on which one may find a non-linear injection preserving the (unique) semi-inner products.