Projections on Invariant Subspaces
and the arguments x are understood to be real numbers modulo 27. It is clear that H' is a closed subspace of Ll. D. J. Newman has shown quite recently [3] that there exist no bounded projections of L' onto H1. (By a projection we mean a linear operator P such that P2= P.) Certain facts about H' suggest a rather general setting for this type of theorem. The abstract problem to which one is thus led turns out to be quite easy, and, when specialized to LI, yields a result which is much stronger than Newman's (Theorem 2). Three facts about H' are relevant. First, H' is translation-invariant. That is to say, if the translation operators r, are defined by