For every element x** in the double dual of a separable Banach space X there exists the sequence $\left(x^{\left(2n\right)}\right)$ of the canonical reproductions of x** in the even-order duals of X. In this paper we prove that every such sequence defines a spreading model for X. Using this result we characterize the elements of X**╲ X which belong to the class $B_1\left(X\right)╲B_{1/2}\left(X\right)$ (resp. to the class $B_{1/4}\left(X\right)$) as the elements with the sequence $\left(x^{\left(2n\right)}\right)$ equivalent to the usual basis of ${\ell }^1$ (resp. as the elements with the sequence $(x^{(4n-2)}-x^{\left(4n\right)})$ equivalent to the usual basis...

We define the radius of the inscribed and circumscribed circumferences in a triangle located in a real normed space and we obtain new characterizations of inner product spaces.

In this note, we prove that a real or complex Banach space $X$ is an $L^1$-predual space if and only if every four-point subset of $X$ is centerable. The real case sharpens Rao’s result in [Chebyshev centers and centerable sets, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2593–2598] and the complex case is closely related to the characterizations of $L^1$-predual spaces by Lima [Complex Banach spaces whose duals are $L_1$-spaces, Israel J. Math. 24 (1976), no. 1, 59–72].

Rosenthal in [11] proved that if $\left(f_k\right)$ is a uniformly bounded sequence of real-valued functions which has no pointwise converging subsequence then $\left(f_k\right)$ has a subsequence which is equivalent to the unit basis of $l^1$ in the supremum norm. Kechris and Louveau in [6] classified the pointwise convergent sequences of continuous real-valued functions, which are defined on a compact metric space, by the aid of a countable ordinal index “$\gamma$”. In this paper we prove some local analogues of the above Rosenthal ’s theorem...

We give a basic sequence characterization of relative weak compactness in c₀ and we construct new examples of closed, bounded, convex subsets of c₀ failing the fixed point property for nonexpansive self-maps. Combining these results, we derive the following characterization of weak compactness for closed, bounded, convex subsets C of c₀: such a C is weakly compact if and only if all of its closed, convex, nonempty subsets have the fixed point property for nonexpansive mappings.

We characterize those anisotropic Sobolev spaces on tori in the $L^1$ and uniform norms for which the idempotent multipliers have a description in terms of the coset ring of the dual group. These results are deduced from more general theorems concerning invariant projections on vector-valued function spaces on tori. This paper is a continuation of the author’s earlier paper [W].

We give a full characterization of the closed one-codimensional subspaces of $c_0$, in which every bounded set has a Chebyshev center. It turns out that one can consider equivalently only finite sets (even only three-point sets) in our case, but not in general. Such hyperplanes are exactly those which are either proximinal or norm-one complemented.

We apply the Chebyshev coefficients λf and λb, recently introduced by the authors, to obtain some results related to certain geometric properties of Banach spaces. We prove that a real normed space E is an L1-predual if and only if λf(E) = 1/2, and that if a (real or complex) normed space E is a P1 space, then λb(E) equals λb(K), where K is the ground field of E.

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of “symmetric exponential” processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity ${\Gamma }_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates...

The study of circumcenters in different types of triangles in real normed spaces gives new characterizations of inner product spaces.

We give two examples of scattered compact spaces K such that C(K) is not uniformly homeomorphic to any subset of c₀(Γ) for any set Γ. The first one is [0,ω₁] and hence it has the smallest possible cardinality, the other one has the smallest possible height ω₀ + 1.

We prove some multi-dimensional Clarkson type inequalities for Banach spaces. The exact relations between such inequalities and the concepts of type and cotype are shown, which gives a conclusion in an extended setting to M. Milman's observation on Clarkson's inequalities and type. A similar investigation conceming the close connection between random Clarkson inequality and the corresponding concepts of type and cotype is also included. The obtained results complement, unify and generalize several...