THE WIGNER PROPERTY OF SMOOTH NORMED SPACES

Abstract

We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping $f: X\rightarrow Y$ satisfying

$$ \begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$

where $\mathbb {T}$ is the unit circle of the complex plane, there exists a function $\sigma : X\rightarrow \mathbb {T}$ such that $\sigma \cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.

1. Introduction

Let X and Y be normed spaces over $\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$ , where $\mathbb {R}$ and $\mathbb {C}$ are the fields of real and complex numbers, respectively. Denote $\mathbb {T}=\{\alpha \in \mathbb {F}:|\alpha |=1\}$ . A function $\sigma :X\rightarrow \mathbb {T}$ whose values are of modulus one is called a phase function on X. A mapping ${f: X \rightarrow Y}$ is said to be phase equivalent to another mapping $g: X \rightarrow Y$ if there exists a phase function $\sigma :X\rightarrow \mathbb {T}$ such that $f=\sigma \cdot g$ , that is, $f(x)=\sigma (x)g(x)$ for $x\in X$ .

The celebrated Wigner’s unitary–anti-unitary theorem is particularly important in the mathematical foundations of quantum mechanics. It states that for inner product spaces $(X,\langle \cdot ,\cdot \rangle )$ and $(Y,\langle \cdot ,\cdot \rangle )$ over $\mathbb {F}$ , a mapping $f: X \rightarrow Y$ satisfies

(1.1) $$ \begin{align} |\langle f(x), f(y) \rangle | = |\langle x, y \rangle|, \quad x, y\in X \end{align} $$

if and only if f is phase equivalent to a linear or anti-linear isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear isometry in the case $\mathbb {F}=\mathbb {R}$ . There are several proofs of this result, see [1, 2, 4, 6, 13, 18, 22] to list just some of them. For further generalisations of this fundamental result, we mention the papers [3, 5, 15, 17]. Wigner’s theorem is very important and therefore worthy of study from various points of view.

A mapping $f: X \rightarrow Y$ between normed spaces over $\mathbb {F}$ is called a phase-isometry if it satisfies the functional equation

(1.2) $$ \begin{align} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X. \end{align} $$

It is worth noting that if X and Y are inner product spaces, then $f: X \rightarrow Y$ satisfies (1.1) if and only if f satisfies (1.2). Indeed, with the substitution $y=x$ , we deduce from either (1.1) or (1.2) that f is norm-preserving. Squaring the norms on both sides of (1.2), it follows that (1.2) holds if and only if

$$ \begin{align*} \{\mbox{Re}(\alpha\langle f(x),f(y) \rangle): \alpha\in \mathbb{T}\}=\{\mbox{Re}(\alpha\langle x, y \rangle): \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$

which happens if and only if (1.1) holds. Due to Wigner’s theorem, a mapping between inner product spaces is a phase-isometry if and only if it is phase equivalent to a linear or anti-linear isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear isometry in the case $\mathbb {F}=\mathbb {R}$ .

When X and Y are normed spaces, one can easily see that if $f:X \rightarrow Y$ is phase equivalent to a linear or anti-linear isometry, then f is a phase-isometry. For instance, if $f=\sigma \cdot U$ , where U is a linear isometry and $\sigma :X\rightarrow \mathbb {T}$ is a phase function, then for $x,y\in X$ and $\alpha \in \mathbb {T}$ ,

$$ \begin{align*} \|f(x)+\alpha f(y)\|&=\|\sigma(x)U(x)+\alpha\sigma(y)U(y)\|=\|U(\sigma(x)x+\alpha\sigma(y)y)\|\\ &=\|\sigma(x)x+\alpha\sigma(y)y\|=\|x+\alpha\overline{\sigma(x)}\sigma(y)y\| \end{align*} $$

and then

$$ \begin{align*} \|x+\alpha y\|=\|x+(\alpha\sigma(x)\overline{\sigma(y)})\overline{\sigma(x)}\sigma(y)y\|=\|f(x)+\alpha\sigma(x)\overline{\sigma(y)} f(y)\|. \end{align*} $$

Similar reasoning applies when U is an anti-linear isometry. Therefore, a natural problem posed by Maksa and Páles [13] (the case $\mathbb {F}=\mathbb {R}$ ), and Wang and Bugajewski [23] (the case $\mathbb {F}=\mathbb {C}$ ), can be restated as the following problem.

Problem 1.1. Under what conditions is every phase-isometry between two normed spaces over $\mathbb {F}$ phase equivalent to a linear or anti-linear isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear isometry in the case $\mathbb {F}=\mathbb {R}$ ?

A normed space X over $\mathbb {F}$ is said to have the Wigner property if for any normed space Y over $\mathbb {F}$ , every surjective phase-isometry $f: X \rightarrow Y$ is phase equivalent to a linear or anti-linear isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear isometry in the case $\mathbb {F}=\mathbb {R}$ .

There have been several recent papers considering Problem 1.1 or the Wigner property in the case $\mathbb {F}=\mathbb {R}$ . For relevant results, please refer to [7–9, 11–13, 19–21, 23]. In particular, Tan and Huang [19] proved that smooth real normed spaces have the Wigner property. Further, Ilišević et al. [9] proved that any real normed spaces have the Wigner property. However, to the best of our knowledge, apart from the case where X and Y are inner product spaces, there has been no progress in addressing Problem 1.1 in the case $\mathbb {F}=\mathbb {C}$ . The aim of this paper is to give a partial solution for the case $\mathbb {F}=\mathbb {C}$ . Specifically, we show that every smooth complex normed space has the Wigner property. As a by-product, we give a Figiel-type result for phase-isometries. Although our paper is interesting in its own right, we hope that it will serve as a stepping stone to show that all complex normed spaces have the Wigner property.

2. Results

In the remainder of this paper, unless otherwise specified, all the normed spaces are over $\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$ . Although the real case has been solved, for the sake of brevity and universality, we will present our lemmas, theorems and proofs in the united form $\mathbb {F}$ rather than the single form $\mathbb {C}$ . For a normed space X, we use the notation $S_X, B_X$ and $X^*$ to represent the unit sphere, closed unit ball and dual space of X, respectively. The set of positive integers is denoted by $\mathbb {N}$ .

We start this section with a simple and frequently-used property of phase-isometries between two normed spaces.

Lemma 2.1. Let X and Y be normed spaces and $f: X \rightarrow Y$ a phase-isometry. Then f is a norm-preserving map. Moreover, if f is surjective, then

$$ \begin{align*} \{f(\alpha x): \alpha\in \mathbb{T}\}=\{\alpha f(x): \alpha\in \mathbb{T}\}, \quad x\in X. \end{align*} $$

Proof. With the substitution $y=x$ , it follows from (1.2) that

$$ \begin{align*} 2\|f(x)\|=\max\{\|f(x)+\alpha f(x)\|: \alpha\in \mathbb{T}\} =\max\{\|x+\alpha x\|: \alpha\in \mathbb{T}\}=2\|x\|, \end{align*} $$

which shows that f is norm-preserving.

Now suppose that f is surjective. Let us take a nonzero $x\in X$ and $\alpha \in \mathbb {T}$ . The surjectivity guarantees that there exists some $y\in X$ such that $f(y)=\alpha f(x)$ . Then

$$ \begin{align*} \min\{\|y+\beta x\|: \beta\in \mathbb{T}\}=\min\{\|f(y)+\beta f(x)\|: \beta\in \mathbb{T}\}=0, \end{align*} $$

which implies that

$$ \begin{align*} \{\alpha f(x): \alpha\in \mathbb{T}\}\subset\{f(\alpha x): \alpha\in \mathbb{T}\}. \end{align*} $$

Moreover, we conclude from (1.2) that

$$ \begin{align*} \min\{\|f(\alpha x)+\beta f(x)\|: \beta\in \mathbb{T}\}=\min\{\|\alpha x+\beta x\|: \beta\in \mathbb{T}\}=0, \end{align*} $$

which shows that

$$ \begin{align*} \{f(\alpha x): \alpha\in \mathbb{T}\}\subset\{\alpha f(x): \alpha\in \mathbb{T}\}. \end{align*} $$

This competes the proof.

From [19, Lemma 2], it follows that every surjective phase-isometry between two real normed spaces is injective. The following example shows that a surjective phase-isometry between two complex normed spaces may not be injective.

Example 2.2. Let X be a complex normed space and $x_0\in X\backslash \{0\}$ . Define ${f:X\rightarrow X}$ by $f(\alpha x_0)=\alpha ^2x_0$ for all $\alpha \in \mathbb {T}$ and $f(x)=x$ otherwise. Then f is a surjective phase-isometry, but it is not injective since $f(-x_0)=x_0=f(x_0)$ .

In Example 2.2, f is phase equivalent to the identity mapping, letting the phase function $\sigma $ be $\sigma (\alpha x_0)=\alpha $ for all $\alpha \in \mathbb {T}$ and $\sigma (x)=1$ otherwise.

Recall that a support functional $\phi $ at $x\in X\backslash \{0\}$ is a norm-one linear functional in $X^*$ such that $\phi (x)=\|x\|$ . Denote by $D(x)$ the set of all support functionals at $x\neq 0$ , that is,

$$ \begin{align*} D(x)=\{\phi\in S_{X^*}: \phi(x)=\|x\|\}. \end{align*} $$

The Hahn–Banach theorem implies that $D(x)\neq \emptyset $ for every $x\in X\backslash \{0\}$ . A normed space X is said to be smooth at $x\neq 0$ if there exists a unique supporting functional at x, that is, $D(x)$ consists of only one element. If X is smooth at every $x\neq 0$ , then X is said to be smooth. It follows from [14, Proposition 5.4.20] that each subspace of a smooth normed space is smooth.

Recall also the concept of Gateaux differentiability. Let X be a normed space, ${x, y\in X}$ . Define

$$ \begin{align*} G_+(x,y):=\lim_{t\to0^+, t\in\mathbb{R}}\frac{\|x+ty\|-\|x\|}{t}=\lim_{t\to+\infty, t\in\mathbb{R}}(\|tx+y\|-\|tx\|) \end{align*} $$

and

$$ \begin{align*} G_-(x,y):=\lim_{t\to0^-, t\in\mathbb{R}}\frac{\|x+ty\|-\|x\|}{t}=\lim_{t\to+\infty, t\in\mathbb{R}}(\|tx\|-\|tx-y\|). \end{align*} $$

It is known [14, 16] that both $G_+(x,y)$ and $G_-(x,y)$ exist for each $x,y\in X$ and

$$ \begin{align*} G_+(x,y)=\max\{\mbox{Re}\,\phi(y):\phi\in D(x)\},\quad G_-(x,y)=\min\{\mbox{Re}\,\phi(y):\phi\in D(x)\}. \end{align*} $$

We say that the norm of X is Gateaux differentiable at $x\neq 0$ whenever $G_+(x,y)=G_-(x,y)$ for all $y\in X$ , in which case the common value of $G_+(x,y)$ and $G_-(x,y)$ is denoted by $G(x,y)$ . It is easy to see that a normed space X is smooth at x if and only if the norm is Gateaux differentiable at x.

A point $\phi \in S_{X^*}$ is said to be a $w^*$ -exposed point of $B_{X^*}$ provided that $\phi $ is the only supporting functional for some smooth point $u\in S_X$ . Recently, Tan and Huang [19] showed that for every phase-isometry f of a real normed space X into another real normed space Y and every $w^*$ -exposed point $\phi $ of $B_{X^*}$ , there exists $\varphi \in S_{Y^*}$ such that $\phi (x)=\pm \varphi (f(x))$ for all $x\in X$ . This result can be viewed as an extension of Figiel’s theorem, which plays an important role in the study of isometric embedding. We will present a similar result for a phase-isometry between two normed spaces over ${\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}}$ .

Lemma 2.3. Let X and Y be normed spaces and $f:X\rightarrow Y$ a phase-isometry. Then for every $w^*$ -exposed point $\phi $ of $B_{X^*}$ , there exists $\varphi \in S_{Y^*}$ such that

$$ \begin{align*} |\phi(x)|=|\varphi(f(x))|, \quad x\in X. \end{align*} $$

Proof. Let $u\in S_X$ be a smooth point such that $\phi (u)=1$ . For every $n\in \mathbb {N}$ , the Hahn–Banach theorem guarantees the existence of $\varphi _n\in S_{Y^*}$ such that

$$ \begin{align*}\varphi_n(f(nu))=\|f(nu)\|= \|nu\|=n.\end{align*} $$

For $t\in [0,n]$ , there exists some $\alpha _{t,n}\in \mathbb {T}$ such that

$$ \begin{align*}\|f(nu)-\alpha_{t,n}f(tu)\|=\|nu-tu\|=n-t.\end{align*} $$

Consequently, we deduce that

$$ \begin{align*} 2n&=|\varphi_n (f(nu)-\alpha_{t,n}f(tu))+\varphi_n(f(nu)+\alpha_{t,n}f(tu))|\\ &\leq|\varphi_n (f(nu)-\alpha_{t,n}f(tu))|+|\varphi_n(f(nu)+\alpha_{t,n}f(tu))|\\ &\leq\|f(nu)-\alpha_{t,n}f(tu)\|+\|f(nu)+\alpha_{t,n}f(tu)\|\\&\leq (n-t)+(n+t)=2n, \end{align*} $$

which implies that $\varphi _n(\alpha _{t,n}f(tu))=t$ . This means that for each $t\in (0, n]$ , there exists a unique $\alpha _{t,n}\in \mathbb {T}$ such that $\varphi _n(f(tu))=\overline {\alpha _{t,n}}t$ . By Alaoglu’s theorem, the sequence $\{\varphi _n\}$ has a cluster point $\varphi \in S_{Y^*}$ in the $w^*$ topology. It follows that for each $t>0$ , there exists $\alpha _t\in \mathbb {T}$ depending only on t such that $\varphi (f(tu))=\alpha _t t$ .

For each $x\in X$ , there exist $\alpha _x, \beta _x\in \mathbb {T}$ such that $\alpha _x\phi (x)=|\phi (x)|$ and $\beta _x\varphi (f(x))=|\varphi (f(x))|$ . For each $n\in \mathbb {N}$ , there exists $\alpha _{x,n}, \beta _{x,n}\in \mathbb {T}$ such that

$$ \begin{align*} \|nu-\alpha_xx\|=\|f(nu)-\alpha_{x,n}\alpha_nf(x)\| & \geq |\varphi(f(nu))-\alpha_{x,n}\alpha_n\varphi(f(x))|\\ &=|\alpha_n n-\alpha_{x,n}\alpha_n\varphi(f(x))|=|n-\alpha_{x,n}\varphi(f(x))| \end{align*} $$

and

$$ \begin{align*} |n+\beta_x\varphi(f(x))|=|\alpha_n n+\alpha_n\beta_x\varphi(f(x))| & =|\varphi(f(nu))+\alpha_n\beta_x \varphi(f(x))|\\ &\leq \|f(nu)+\alpha_n\beta_xf(x)\|=\| nu+\beta_{x,n}x\|. \end{align*} $$

Given that $\mathbb {T}$ is compact, there must be a strictly increasing sequence $\{n_j:j\in \mathbb {N}\}$ in $\mathbb {N}$ and $\alpha ^{\prime }_x,\beta ^{\prime }_x\in \mathbb {T}$ such that $\lim _{j\to \infty }\alpha _{x,n_j}=\alpha ^{\prime }_x$ and $\lim _{j\to \infty }\beta _{x,n_j}=\beta ^{\prime }_x$ . Since $\phi $ is the only supporting functional at u,

$$ \begin{align*} |\phi(x)|&=\mbox{Re}\,\phi(\alpha_xx)=\lim_{j\to\infty}(\|n_ju\|-\|n_ju-\alpha_xx\|)\\ &\leq\lim_{j\to\infty}(n_j-| n_j-\alpha_{x,n_j}\varphi(f(x))|)=\lim_{j\to\infty}(n_j-| n_j-\alpha^{\prime}_x\varphi(f(x))|)\\ &=\mbox{Re}\,(\alpha^{\prime}_x\varphi(f(x)))\leq|\varphi(f(x))| \end{align*} $$

and

$$ \begin{align*} |\varphi(f(x))|&=\mbox{Re}\,(\beta_x\varphi(f(x)))=\lim_{j\to\infty}(|n_j+\beta_x\varphi(f(x))|-n_j)\\ &\leq\lim_{j\to\infty} (\| n_ju+\beta_{x,n_j}x\|-\|n_ju\|)=\lim_{j\to\infty} (\|n_ju+\beta^{\prime}_xx\|-\|n_ju\|)\\ &=\mbox{Re}\,\phi(\beta^{\prime}_xx)\leq|\phi(x)|. \end{align*} $$

This completes the proof.

Let V be a vector space. For $M\subset V$ , $[M]$ denotes the subspace generated by M. If $x,y\in V$ , then we write $[x]:=[\{x\}]$ and $[x,y]:=[\{x,y\}]$ for simplicity.

Lemma 2.4. Let X and Y be normed spaces with X being smooth. Suppose that ${f: X \rightarrow Y}$ is a surjective phase-isometry. Then for every $x\in X$ ,

$$ \begin{align*} f([x])=[f(x)]. \end{align*} $$

Proof. We first prove that $[f(x)]\subset f([x])$ for each $x\in X$ . Assume, for a contradiction, that $tf(x)\notin f([x])$ for some nonzero $x\in X$ and $t\in \mathbb {F}$ . Since f is surjective, there exists $y\in X$ such that $f(y)=tf(x)$ . The function $s\mapsto \|y-sx\|$ is continuous and its value tends to infinity when $|s|$ tends to infinity. Hence, there is at least one point $s_0\in \mathbb {F}$ such that

$$ \begin{align*} d:=d(y,[x])=\min\{\|y-sx\|: s\in \mathbb{F}\}=\|y-s_0x\|>0. \end{align*} $$

Set $E:=[x,y]$ . By the Hahn–Banach theorem, there exists $\phi \in S_{E^*}$ which satisfies $\phi (y)=d$ and $\phi (x)=0$ . Note that E being a two-dimensional subspace of X is reflexive. This guarantees the existence of some $z\in S_E$ such that $\phi (z)=1$ . Since X is smooth, so is its subspace E. Therefore, $\phi $ is the only supporting functional at $z\in S_E$ . We apply Lemma 2.3 to $f|_E: E\rightarrow Y$ to obtain $\varphi \in S_{Y^*}$ such that $|\phi |=|\varphi \circ f|$ on E. Then

$$ \begin{align*} 0<d=|\phi(y)|=|\varphi(f(y))|=|\varphi(tf(x))|=|t||\varphi(f(x))|=|t||\phi(x)|=0, \end{align*} $$

which is a contradiction. This proves $[f(x)]\subset f([x])$ .

Conversely, fix a nonzero $x\in X$ . For each $r\in (0,+\infty )$ , by the above inclusion and the norm preserving property of f, there exists some $\alpha _r\in \mathbb {T}$ such that $r^{-1}f(rx)=f(\alpha _r x)$ . For each $\alpha \in \mathbb {T}$ , by Lemma 2.1, there exist $\beta _{r,\alpha },\alpha ^{\prime }_r\in \mathbb {T}$ such that

$$ \begin{align*} f(r\alpha x)=\beta_{r,\alpha}f(rx)=\beta_{r,\alpha}rf(\alpha_r x)=r\beta_{r,\alpha}\alpha^{\prime}_rf(x), \end{align*} $$

which implies that $f([x])\subset [f(x)]$ . The proof is complete.

Note that the conclusion of Lemma 2.4 is equivalent to

$$ \begin{align*} \{f(r\alpha x): \alpha\in \mathbb{T}\}=\{r\alpha f(x): \alpha\in \mathbb{T}\}, \quad x\in X,\ r\in [0,+\infty). \end{align*} $$

Lemma 2.5. Let X and Y be normed spaces with X being smooth. Suppose that ${f: X \rightarrow Y}$ is a surjective phase-isometry. Then for every $x,y\in X$ ,

$$ \begin{align*} \{G_+(f(x),\alpha f(y)): \alpha\in \mathbb{T}\}=\{G(x,\alpha y): \alpha\in \mathbb{T}\}=\{G_-(f(x),\alpha f(y)): \alpha\in \mathbb{T}\}. \end{align*} $$

Proof. We only prove the first equality, the second being similar. Let $x, y\in X$ be nonzero and $\alpha \in \mathbb {T}$ . For each $n\in \mathbb {N}$ , Lemma 2.4 and (1.2) imply that there exist $\alpha _n,\beta _n,\gamma _n\in \mathbb {T}$ such that $f(nx)=\alpha _nnf(x)$ and

$$ \begin{align*} \|f(nx)+\alpha_n\alpha f(y)\|=\|nx+\beta_n y\|, \quad \|f(nx)+\alpha_n\gamma_nf(y)\|=\|nx+\alpha y\|. \end{align*} $$

By the compactness of $\mathbb {T}$ , there is a strictly increasing sequence $\{n_j:j\in \mathbb {N}\}$ in $\mathbb {N}$ and $\beta ,\gamma \in \mathbb {T}$ such that $\lim _{j\to \infty }\beta _{n_j}=\beta $ and $\lim _{j\to \infty }\gamma _{n_j}=\gamma $ . Then

$$ \begin{align*} G_+(f(x),\alpha f(y)) & =\lim_{j\to\infty}(\|n_jf(x)+\alpha f(y)\|-\|n_jf(x)\|)\\ & = \lim_{j\to\infty}(\|f(n_jx)+\alpha_{n_j}\alpha f(y)\|-\|n_jf(x)\|) \\ &= \lim_{j\to\infty}(\|n_jx+\beta_{n_j}y\|-\|n_jx\|) = \lim_{j\to\infty}(\|n_jx+\beta y\|-\|n_jx\|)=G(x,\beta y) \end{align*} $$

and

$$ \begin{align*} G(x,\alpha y)&=\lim_{j\to\infty}(\|n_jx+\alpha y\|-\|n_jx\|)\\ &=\lim_{j\to\infty}(\|f(n_jx)+\alpha_{n_j}\gamma_{n_j}f(y)\|-\|f(n_jx)\|)\\ &=\lim_{j\to\infty}(\|n_jf(x)+\gamma_{n_j}f(y)\|-\|n_jf(x)\|)\\ &=\lim_{j\to\infty}(\|n_jf(x)+\gamma f(y)\|-\|n_jf(x)\|)=G_+(f(x),\gamma f(y)). \end{align*} $$

The proof is complete.

Lemma 2.6. Let X and Y be normed spaces with X being smooth. Suppose that ${f: X \rightarrow Y}$ is a surjective phase-isometry. Then Y is smooth.

Proof. Let $x\in X$ be a nonzero element with the unique supporting functional ${\phi _x\in D(x)}$ . It suffices to prove that $D(f(x))$ is a singleton set. Let $\varphi ,\psi \in D(f(x))$ and $f(y)\in \ker \varphi $ . For each $\alpha \in \mathbb T$ , Lemma 2.5 implies that there exists $\beta ,\gamma \in \mathbb T$ such that

$$ \begin{align*} \mbox{Re}(\alpha \phi_{x}(y))=\mbox{Re}\phi_{x}(\alpha y)=G(x,\alpha y)=G_+(f(x),\beta f(y))\geq\mbox{Re}\varphi(\beta f(y))=0 \end{align*} $$

and

$$ \begin{align*} \mbox{Re}(\alpha\psi(f(y)))=\mbox{Re}\psi(\alpha f(y))\leq G_+(f(x),\alpha f(y))=G(x,\gamma y)=\mbox{Re} \phi_{x}(\gamma y). \end{align*} $$

Using the arbitrariness of $\alpha \in \mathbb T$ twice gives $\phi _{x}(y)=0$ by the first inequality and therefore $\psi (f(y))=0$ by the second inequality. This shows that $\ker \varphi \subset \ker \psi $ . Thus, $\psi =\lambda \varphi $ for some $\lambda \in \mathbb F$ . Considering that $\psi ,\varphi \in D(f(x))$ , we find that $\lambda =1$ . This implies that $\psi =\varphi $ , which completes the proof.

Recently, Ilišević and Turnšek [10, Theorem 2.2 and Remark 2.1] generalised Wigner’s theorem to smooth normed spaces via semi-inner products. This can be translated into the following theorem in the language of supporting functionals.

Theorem 2.7. Let X and Y be smooth normed spaces over $\mathbb {F}$ and $f:X\rightarrow Y$ a surjective mapping satisfying, for all nonzero $x, y\in X$ ,

$$ \begin{align*} |\phi_{f(x)}(f(y))| = |\phi_x(y)|. \end{align*} $$

Then f is phase equivalent to a linear or anti-linear surjective isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear surjective isometry in the case $\mathbb {F}=\mathbb {R}$ .

Combining the above results gives our main theorem.

Theorem 2.8. Every smooth normed space has the Wigner property.

Proof. Let X and Y be normed spaces with X being smooth. Suppose that $f: X \rightarrow Y$ is a surjective phase-isometry. By Lemma 2.6, Y is smooth. Then Lemma 2.5 implies that for all nonzero $x, y\in X$ ,

$$ \begin{align*} \{\mbox{Re}\phi_{f(x)}(\alpha f(y)): \alpha\in \mathbb{T}\}=\{\mbox{Re}\phi_{x}(\alpha y): \alpha\in \mathbb{T}\}. \end{align*} $$

Taking the maximum on both sides, for all nonzero $x, y\in X$ ,

$$ \begin{align*} |\phi_{f(x)}(f(y))|=|\phi_{x}(y)|. \end{align*} $$

By Theorem 2.7, f is phase equivalent to a linear or anti-linear surjective isometry in the case $\mathbb {F}=\mathbb {C}$ and to a linear surjective isometry in the case $\mathbb {F}=\mathbb {R}$ . This completes the proof.

It is well known that $L^p(\mu )$ is a smooth normed space, where $\mu $ is a measure and $1<p<\infty $ . The following corollary is immediate.

Corollary 2.9. $L^{p}(\mu )$ has the Wigner property, where $\mu $ is a measure and ${1<p<\infty }$ .