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A number r > 0 is called a rendezvous number for a metric space (M, d) if for any n ∈ ℕ and any x1,…xn ∈ M, there exists x ∈ M such that . A rendezvous number for a normed space X is a rendezvous number for its unit sphere. A surprising theorem due to O. Gross states that every finite dimensional normed space has one and only one average number, denoted by r (X). In a recent paper, A. Hinrichs solves a conjecture raised by R. Wolf. He proves that for any n-dimensional real normed space. In this paper, we prove the analogous inequality in the complex case for n ≥ 3.

Let X be an infinite dimensional Banach space. The paper proves the non-coincidence of the vector-valued Hardy space Hp([ ], X) with neither the projective nor the injective tensor product of Hp([ ]) and X, for 1 < p < ∞. The same result is proved for some other subspaces of Lp. A characterization is given of when every approximable operator from X into a Banach space of measurable functions [Fscr ](S) is representable by a function F:S → X as x [map ] 〈F(·), x〉. As a consequence the existence is proved of compact operators from X into Hp([ ]) (1 [les ] p < ∞) which are not representable. An analytic Pettis integrable function F:[ ] → X is constructed whose Poisson integral does not converge pointwise.

In this note we calculate the exact values of the rendezvous numbers of the Banach spaces $ \ell _\infty ^n({\Bbb C}) $ and $ \left({1\over 3}+{{2\sqrt 3}\over{\pi }}\right) $ , and $ \ell _1^n({\Bbb C}) $ (given in terms of the complete elliptic integral function). This answer some questions appeared in [5]. We also consider the space $ \scri C (K,{\Bbb C}) $ of all $ {\Bbb C} $ -vlued, continous functions defined on a Hausdorff compact set K. We prove that if K has no isolated point the every $ \alpha \in {\left({1\over 3}+{{2\sqrt 3}\over{\pi }},2\right)} $ is a rendezvous number of $ \scr C (K,{\Bbb C}) $ , and if K has at least one isolated point then $ {1\over 3}+{{2\sqrt 3}\over{\pi }} $ is the unique rendezvous number of $ \scr C (K,{\Bbb C}) $ .

Some characterizations of integrable functions in the bilinear sense of Bartle with respect to the injective tensor product are obtained. As a consequence it is shown that the kernels of Carleman compact operators coincide with these Bartle integrable functions. This result is applied to prove that every Carleman L-weak-compact operator is compact. An example showing the different behavior of the integrability with respect to the projective tensor product is given. A general Fubini theorem in this setting is shown.

In this paper, it is proved that, for any m unit vectors. x 1 …, x m in any n -dimensional real Hilbert space, there exists a unit vector x 0 such that for any y ∈ S n −1 . The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.

We present some recent results on operators with values in Hardy spaces and vector valued functions. Some compact operators are constructed which are not representable by functions and the non coincidence of the vector valued Hardy spaces with either the projective or the injective tensor product is proved. We improve some previous results on the failure of Fatou’s theorem on radial almost everywhere convergence for analytic Pettis integrable functions.

In this paper we prove that for every infinite-dimensional Banach spaceXand every 1⩽p

We characterize those Banach spaces X X , in which every X X -valued measure with relatively compact range admits product with any vector measure and with respect to any bilinear map, as those X X such that Π 1 ( X , ℓ 1 ) = L ( X , ℓ 1 ) \Pi _{1} (X,\ell _{1}) = {\mathcal {L}} (X,\ell _{1}) . We also show that this condition is equivalent to the condition that every sequence in X X that lies inside the range of a measure with relatively compact range, actually lies inside the range of a measure of bounded variation.