A subspaceY of a Banach spaceX is called a Chebyshev one if for everyx∈X there exists a unique elementPY(x) inY of best approximation. In this paper, necessary and sufficient conditions are obtained in order that certain classes of subspacesY of the Hardy spaceH1=H1 (|z|<1) be Chebyshev ones, and also the properties of the operatorPY are studied. These results show that the theory of Chebyshev subspaces inH1 differs sharply from the corresponding theory inL1(C) of complex-valued functions defined and integrable on the unit circleC:|z|=1. For example, it is proved that inH1 there exist sufficiently many Chebyshev subspaces of finite dimension or co-dimension (while inL1(C) there are no Chebyshev subspaces of finite dimension or co-dimension). Besides, it turned out that the collection of the
Chebyshev subspacesY with a linear operatorPY inH1 (in contrast toL1(C)) is exhausted by that minimum which is necessary for any Banach space.
geometrical conditions that implies the existence certain singular integral of Banachspace-valued functions
Proc. Conf. Harmonic Analysis
Chicago 1981 Wads Worth Belmont 270 – 286 in honor of Antoni Zygmund
The tensor integral of a vector valued function f: Ω → X with respect to a countably additive vector valued measure v: Σ → Y has been defined by Stefansson in  and he has investigated many of its properties. The integral is an element of the
injective tensor product X
of double sequences whose Cesàro transforms are bounded, convergent in the Pringsheim’s sense, null in the Pringsheim’s sense, both convergent in the Pringsheim’s sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space