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.
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