In this note we generalize the well-known Wigner's unitary-anti\-unitary theorem. For $X$ and $Y$ smooth normed spaces and $f:X\to Y$ a surjective mapping such that $|[f(x),f(y)]|=|[x,y]|$, $x,y\in X$, where $[\cdot,\cdot]$ is the unique semi-inner product, we show that $f$ is phase equivalent to either a linear or an anti-linear surjective isometry. When $X$ and $Y$ are smooth real normed spaces and $Y$ strictly convex, we show that Wigner's theorem is equivalent to $\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\{\|x+y\|,\|x-y\|\}$, $x,y\in X$.
We generalize the notion of $M$-ideals in order smooth $\infty$-normed spaces to "smooth $p$-order ideals" in order smooth $p$-normed spaces. We show that if $V$ is an order smooth $p$-normed space and $W$ is a closed subspace of $V$, then $W$ is a smooth $p$-order ideal in $V$ if and only if $W^{\perp}$ is a smooth $p'$-order ideal in order smooth $p'$-normed space if and only if $W^{\perp\perp}$ is a smooth $p$-order ideal in order smooth $p$-normed space $V^{**}$. We prove that every $L$-summand in order smooth $1$-normed space is a smooth $1$-order ideal. We find a condition under which ev…
We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.
Let $X$ and $Y$ be real normed spaces and $f \colon X\to Y$ a surjective mapping. Then $f$ satisfies $\{\|f(x)+f(y)\|, \|f(x)-f(y)\|\} = \{\|x+y\|, \|x-y\|\}$, $x,y\in X$, if and only if $f$ is phase equivalent to a surjective linear isometry, that is, $f=σU$, where $U \colon X\to Y$ is a surjective linear isometry and $σ\colon X\to \{-1,1\}$. This is a Wigner's type result for real normed spaces.
We say that a mapping $f: X \rightarrow Y$ between two real normed spaces is a phase-isometry if it satisfies the functional equation \begin{eqnarray*} \{\|f(x)+f(y)\|, \|f(x)-f(y)\|\}=\{\|x+y\|, \|x-y\|\} \quad (x,y\in X).\end{eqnarray*} A generalized Mazur-Ulam question is whether every surjective phase-isometry is a multiplication of a linear isometry and a map with range $\{-1, 1\}$. This assertion is also an extension of a fundamental statement in the mathematical description of quantum mechanics, Wigner's theorem to real normed spaces. In this paper, we show that for every space $Y$ the …
We characterize $M$-ideals in order smooth $\infty$-normed spaces by extending the notion of split faces of the state space to those of the quasi-state space. We also characterize approximate order unit spaces as those order smooth $\infty$-normed spaces $V$ that are $M$-ideals in $\tilde{V}.$ Here $\tilde{V}$ is the order unit space obtained by adjoining an order unit to $V.$ To prove these results, we develop an order theoretic version of the "Alfsen-Efffros' cone decomposition theorem" for order smooth $1$-normed spaces. (As a quick application of this result, we sharpen a result on the ext…
We introduce a notion of p-orthogonality in a general Banach space $1 \le p \le \infty$. We use this concept to characterize $\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of $p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the $\infty$-orthogonal decomposition holds in an order smooth $\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.
The aim of this paper is to present some results concerning the $ρ_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, ρ_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\leq 3[T]\|x\|, \qquad (x\in X)$$ where $[T]:=\inf\{\|Tx\|: \,x\in X, \|x\|=1\}$. We also show that the pair $(X,\perp_{ρ_*})$ is an orthogonality space in the sense of Rätz. Some characterizations of smooth spaces are given based on the $ρ_*$-orthogonality.
We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows by our results that an LUR or a uniformly Fréchet smooth infinite-dimensional Banach space does not admit star-finite coverings by closed balls. On the other hand, we present a quite involved construction proving existence of a star-finite covering of $c_0(Γ)$ by Fréchet smooth centrally symmetric bounded convex bodies. A similar but simpler construction shows that every normed space of countabl…
Jagiellonian University（雅盖隆大学）；Pedagogical University of Cracow（克拉科夫师范大学）
We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space $\ell_p$, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach sp…
An equilateral dimension of a normed space is a maximal number of pairwise equidistant points of this space. The aim of this paper is to study the equilateral dimension of certain classes of finite dimensional normed spaces. The well-known conjecture states that the equilateral dimension of any $n$-dimensional normed space is not less than $n+1$. By using an elementary continuity argument, we establish it in the following classes of spaces: permutation-invariant spaces, Orlicz-Musielak spaces and in one codimensional subspaces of $\ell^n_{\infty}$. For smooth and symmetric spaces, Orlicz-Musie…
We study the sizes of delta-additive sets of unit vectors in a d-dimensional normed space: the sum of any two vectors has norm at most delta. One-additive sets originate in finding upper bounds of vertex degrees of Steiner Minimum Trees in finite dimensional smooth normed spaces (Z. Füredi, J. C. Lagarias, F. Morgan, 1991). We show that the maximum size of a delta-additive set over all normed spaces of dimension d grows exponentially in d for fixed delta>2/3, stays bounded for delta<2/3, and grows linearly at the threshold delta=2/3. Furthermore, the maximum size of a 2/3-additive set in d-dim…
In this paper we establish that if a Tychonoff space $X$ is Čech-complete then the space $O_τ(X)$ of all $τ$-smooth order-preserving, weakly additive and normed functionals is also Čech-complete
We give a rate of metastability for Halpern's iteration relative to a rate of metastability for the resolvent for nonexpansive mappings in uniformly smooth Banach spaces, extracted from a proof due to Xu. In Hilbert space, the latter is known, so we get an explicit rate of metastability. We also extract a rate of asymptotic regularity for general normed spaces. Such rates have already been extracted by Kohlenbach and Leuştean for different and incomparable conditions on $(λ_n)$. The proof analyzed in this paper is also more effective than the proofs treated by Kohlenbach and Leuştean in that i…
The main goal of this paper is to provide an alternative proof of the following theorem of Petty: in the normed space of dimension at least three, every 3-element equilateral set can be extended to a 4-element equilateral set. Our approach is based on the result of Kramer and Németh about inscribing a simplex into a convex body. To prove the theorem of Petty, we shall also establish that for every 3 points in the normed plane, forming an equilateral set of the common distance $p$, there exists a fourth point, which is equidistant to the given points with the distance not larger than $p$. We wi…
