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Let ( M 1 , d 1 ) , ( M 2 , d 2 ) be metric spaces. A map f : M 1 → M 2 is said to be locally open at an x 1 ∈ M 1 , if for every e > 0 one finds a δ > 0 such that B ( f ( x 1 ) , δ ) ⊂ f ( B ( x 1 , e ) ) ; here B ( x , r ) stands for the closed ball with center x and radius r . We are particularly interested in the following special case: X , Y , Z are normed spaces, the spaces L ( X , Y ) , L ( Y , Z ) , L ( X , Z ) of linear continuous operators are provided with the operator norm, and the map under consideration is the bilinear map ( S , T ) ↦ S ∘ T (from L ( Y , Z ) × ( L ( X , Y ) to L ( X , Z ) ). For which pairs ( S 0 , T 0 ) ∈ L ( Y , Z ) × ( L ( X , Y ) is it locally open? The main result of the paper gives a complete characterization of pairs ( S , T ) at which this map is locally open in the case of finite-dimensional spaces X , Y , Z .

1 The phenomenon Let p be a prime number. We write Zp for the set {0, 1, . . . , p−1} of residues modulo p, and we consider the usual addition and multiplication modulo p onZp . It will be important in the sequel that Zp , provided with these operations, is a field, i.e., one calculates as in the usual number system. (E.g., for x = 0, the number 1/x is defined as that element y ∈ Zp for which x · y = 1. In the case p = 7, for example, one has 1/5 = 3. And −x means that y such that x + y = 0. E.g., −1 = p − 1 for x ∈ Zp .) Now let numbers a, b ∈ Zp be given. They generate a sequence x0, x1, . . . in Zp by x0 := a, x1 := b, xn := xn−1 + xn−2 mod p for p ≥ 2. Note that this is the usual Fibonacci sequence modulo p in the case (a, b) = (0, 1).

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces $L^{p}:=L_{\mathbb{K}}^{p}$ (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) and pointwise multiplication from $L^{p}\times L^{q}$ to $L^{r}$ (where $1/p+1/q=1/r$ ). The second result is concerned with the nontrivial $n$ -linear functionals from the product $X_{1}\times\cdots\times X_{n}$ of normed spaces (over $\mathbb{K}\in\{\mathbb{R},\mathbb{C}\}$ ) to the field $\mathbb{K}$ .

We call a sequence (Tn) of measure preserving transformations strongly mixing if \(P(T_n^{ - 1} A \cap B)\) tends to P(A)P(B) for arbitrary measurable A, B. We investigate whether one can pass to a suitable subsequence \(T_{n_k } \) such that \(\frac{1}{K}\sum\limits_{k = 1}^k {f(T_{n_k } )} \to \smallint fdP\) almost surely for all (or "many") integrable f.

A metric space (M;d) is said to have the small ball property (sbp) if for every "0 > 0 it is possible to write M as the union of a sequence (B(xn;rn)) of closed balls such that the rn are smaller than "0 and limrn = 0. We study permanence properties and examples of sbp. The main results of this paper are the following: 1. Bounded convex closed sets in Banach spaces have sbp only if they are compact. 2. Precisely the nite- dimensional Banach spaces have sbp. (More generally: a complete metric group has sbp i it is separable and locally compact.) 3. Let B be a boundary in the bidual of an innite-dimension al Banach space. Then B does not have sbp. In particular the set of extreme points in the unit ball of an innite-dimension al reexiv e Banach space fails to have sbp.

Ehrhard Behrends ist 1946 geboren. Seit 1973 ist er Professor an der Freien Universitat Berlin. Sein Hauptarbeitsgebiet ist die Funktionalanalysis, er hat aber auch schon uber Topologie, Ergodentheorie und Wahrscheinlichkeitsrechnung gearbeitet. Unter seinen Publikationen finden sich vier Bucher. Seine Interessen ausserhalb der Mathematik liegen im Bereich der Musik. Freizeit und Beruf beruhren sich manchmal; so organisierte er am Internationalen Mathematiker-Kongress in Berlin Veranstaltungen zum Thema „Mathematik und Musik“.

Matrix-closed families of subspaces of a Banach space have been of importance in connection with generalizations of the local reflexivity theorem. The present paper provides sufficient conditions for a family of subspaces to be matrix-closed which can easily be checked. Using this result it is, e.g., simple to decide whether a family consisting of three subspaces is matrix-closed or not. It is also shown that this is in a sense the best possible general assertion, since no similar result can hold for the case of four subspaces.

Let $C[0,1]$ be the Banach algebra of real valued continuous functions on $[0,1]$, provided with the supremum norm. For $f,g\in C[0,1]$ and balls $B_{f}$, $B_{g}$ with center $f$ and $g$, respectively, it is not necessarily true that $f\cdot g$ is in the interior of $B_{f}\cdot B_{g}$. In the present paper we characterize those pairs $f,g$ where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.

Let x 1 , l , x n and y 1 , l , y n be vectors in the finite dimensional euclidean space R d . We investigate the problem of how one can find a U such that ∑ i 〈 U x i , y i 〉 is maximal when U runs through the orthogonal group or the special orthogonal group, i.e., we are looking for a U such that the U x 1 , l ,U x n and the y 1 , l , y n are “as parallel as possible”. For d =3 this problem arises, for instance, from the data analysis of crystallographic diffraction experiments on orientationally disordered systems: the x 's stand for the atom positions of fragments M 1 of the crystal structure, the y 's are taken from the set M 2 of maxima of the electron-density distribution reconstructed from diffraction data, and one must know the above transformations U in order to determine the M 1 – M 2 -configurations of minimum distance since they are responsible for the shape of the density distribution. It turns out that one can associate a d × d -matrix G with this problem in such a way that the relevant U are precisely those for which the trace of GU is maximal. Using this transformation we are able to provide an explicit solution by means of the singular value decomposition of G . Several further topics in connection with this problem are also discussed. In particular we investigate in which cases the optimal position is unique, and we study generalizations to nondiscrete situations and to the infinite dimensional setting.

Let $X$ be a connected separable linear order, a connected separable metric space, or a connected, locally connected complete metric space. We show that every continuous function $f:X\to\mathbb R$ with the property that every $x\in X$ is a local maximum or minimum of $f$ is in fact constant. We provide an example of a compact connected linear order $X$ and a continuous function $f:X\to\mathbb R$ that is not constant and yet every point of $X$ is a local minimum or maximum of $f$.