Showing papers on "Birnbaum–Orlicz space published in 1971"

On orlicz sequence spaces

Abstract: It is proved that every Orlicz sequence space contains a subspace isomorphic to somel p . The question of uniqueness of symmetric bases in Orlicz sequence spaces is investigated.

On a class of real Banach spaces

Abstract: The structure theory for simplex spaces is extended to arbitrary real Banach spaces with L1-duals.

Quasi-differentiable functions of Banach spaces

Abstract: Nonzero Frechet differentiable functions with bounded support do not exist on certain real separable Banach spaces. As a result, the class of differentiable functions on such spaces is too small to be useful. For example, the class of differentiable functions on certain spaces does not separate disjoint closed subsets of the space. It is shown that this separation problem does not arise if Frechet differentiability is replaced by the weaker condition of quasi-differentiability. Furthermore, it is shown that any bounded uniformly continuous function on a real separable Banach space is the uniform limit of quasi-differentiable functions.

Seminar on potential theory, II

Abstract: Functional spaces and their exceptional sets.- Dirichlet forms on regular functional spaces.- Cohomology in harmonic spaces.- Martin boundary and ?p-theory of harmonic spaces.- Approximation of capacities by measures.

Continuous function spaces with integral norms

Abstract: There is a long history of utilizing spaces of continuous functions for investigating the structure of topological spaces. A spectacular example is the Banach-Stone Theorem, which asserts tha t two compact Hausdorff spaces T 1 and T~ are homeomorphic if and only if the normed linear spaces C(T1) and C(T2) axe linearly isometric. Another remarkable theorem, closer to the subject of this paper, is tha t of Borsuk and Dugundji : if the topological space T is metrisable then to each closed set Q in T there corresponds a monotone, linear, isometric extension map from C(Q) to C(T). In many results of this kind, the space C(T) is assigned the suprcmum norm, ]1 x Ho~ = sup {Ix(t)]: t e T}. Problems in applied mathematics, in approximation theory, and in statistics, however, often involve a closely-related space C~(T), which is defined to be the linear space C(T) endowed with a norm of the form H x I1~ = ~/f ]x I p. I t is natura l to inquire then to what extent the topological properties of T are reflected in the structure of C~ (T). Taking a broader view, one can s tudy these spaces on their own merits because of their intrisic importance and their frequent occurrence. Not much work in this general direction has come to our attention. The principal exceptions appear to be the approximation-theoretic studies of CI(T) initiated by D. JACKSOn [8] and 3I. G. t~_REIN [10]. La ter investigations by PT.iK [13], KRIPKE and RIVLIN [11], PHELPS [12], HAu [7a], and CHENEu and WULBERT [3] are concerned mainly with the central problem considered already by JACKSOn and ~kREIN, namely, to characterize those linear subspaces in CI(T ) such tha t each member of C1 (T) possesses a unique best approximation in the subspace. Here we continue these investigations and consider other problems in which the topology of T plays a more decisive role. In particular, we focus a t tent ion on linear and nonlinear extension maps and on the existence problem for best approximations in subspaces.

Chebyshev inequalities in symmetric spaces

Abstract: The characterization (by means of inequalities) of some special Banach spaces is investigated.

Conditional expectations and an isomorphic characterization of ₁-spaces

Abstract: Conditional expectations can be defined in Banach spaces whose elements can be represented as measurable fuinctions. In the present paper it is shown that such a space (precisely a cyclic space) is isomorphic to an Li-space if and only if the conditional expectations act as bounded operators for sufficiently many representations. Let (Q, 1, /t) be a finite measure space and 2o a subring of z with maximal element Qo; then for each 1-measurable functionf which is bounded on Q one can consider the measure bAo(o) =fsnsf(co)J1u(dco); oQoEzC2;o. Since /.0 is evidently absolutely continuous with respect to the restriction of /.t to the subfield generated by 2o and Q, due to the Radon-Nikodym theorem, there exists a 20-measurable function denoted E(2o, It)f for which ff(w)IA(dw) = f E(Zo, Iu)fu(dw); of C 2O Obviously the operator E(2o, u) :f-*E(24, f)f can be extended uniquely to a contractive projection in Lp(Q, 1, /u); 1 _p _ + oo, which is called the conditional expectation relative to 10. However, if the L,-norm is replaced by a general monotonic norm p in the sense of the theory of Banach function spaces (see for instance W. A. J. Luxemburg and A. C. Zaanen [9, Note I ]), usually, E(20, bt) does not act as a bounded operator in Lp-the space of all 2-measurable functions for which p(f) < + so, even when we assume that L1(Qi, , DL)DLpDLo(Q, 1, p). Furthermore, a Banach function space Lp admits many isometric representations; e.g. to every positive function EzLp whose support is Q one can define a new norm po(f) =p(of), obtaining in this way a new Banach function space Lpf which is isometric to Lp and satisfies L1(Q, 1, A) DLLpo DLo(Q, 1, OA). The main result of this paper states that Lp is isomorphic to an Li-space over a finite measure space provided for every subring 2o of Received by the editors September 24, 1969. AMS 1969 subject classifications. Primary 4606, 4635, 4725; Secondary 4610, 4720.