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ALMOST CONTRACTIVE RETRACTIONS IN ORLICZ SPACESGRZEGORZ LEWICKI AND GIULIO TROMBETTA Let Bk denote the Euclidean unit ball in R* equipped with the A:-dimensional Lebesgue measure and let : K + -* R + be a convex function satisfying 0(0) = 0, 0 for some t > 0. Denote by E* = E^{B k ) the Orlicz space of finite elements (see (1.6)) generated by

In this paper we consider the Wos' ko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k ^ 1 for which there exists a fc-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct.

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Abstract. We prove the admissibility of the space L 0 (A, X) of vectorvalued measurable functions determined by real-valued finitely additive set functions defined on algebras of sets.The notion of admissibility introduced by Klee [7] guarantees that a compact mapping into an admissible Hausdorff topological vector space E can be approximated by compact finite dimensional mappings. This notion is very important in degree theory and fixed point theory. It is known that locally convex spaces are admissible (see [10]). There are some classes of nonlocally convex spaces which are admissible. Riedrich in [13] proved the admissibility of the space S(0, 1) of measurable functions and in [12] the admissibility of the space L p (0, 1) for 0 < p < 1. The admissibility of other function spaces has been proved by Mach [6] and Ishii [8]. In [14] it is proved the admissibility of spaces of Besov-Triebel-Lizorkin type. Definition 1 ([7]). Let E be a Haudorff topological vector space. A subset Z of E is said to be admissible if for every compact subset K of Z and for every neighborhood V of zero in E there exists a continuous mapping H : K → Z such that dim(span [H(K)])< +∞ and x − Hx ∈ V for every x ∈ K. If Z = E we say that the space E is admissible.In this paper we deal with spaces of vector-valued measurable functions and, as a major fact, instead of σ-additive measures we consider finitely additive set functions defined on algebras of sets.Let X be a Banach space, Ω a nonempty set, A a subalgebra of the power set P(Ω) of Ω and µ : A → R a finitely additive set function. We prove

We construct retractions with positive lower Hausdorff norms and small Hausdorff norms in Banach spaces of real continuous functions which domains are not necessarily bounded or finite dimensional. Moreover, we give precise formulas for the lower Hausdorff norms and the Hausdorff norms of such maps.