The Hardy space Hp=Hp(U), where 1≤p≤∞ and U is the unit disc |z|<1, is shown to be an algebra under the product

Moreover for each p, 1≤p≤∞, there exists a constant Cp such that ‖f*g‖p,≤ Cp‖f‖p ‖g‖p.

In certain systems analysis ([1], [2], [3]), it is essential to invert the n-dimensional Laplace transform and specify the inverse image at a single variable t.

In [5] R. C. Lyndon gave the first proof of the Vaught conjecture: that if a, b9 and c are elements of a free group F such that a2b2=c2, then ab=ba. Lyndon's proof has been followed by many alternative proofs and generalizations [1, 2, 3, 4, 6, 8, 9, 10, 11, 13, 14] all of which involve rather long combinatorial arguments or group theoretical arguments of a noncombinatorial nature.This note provides a short, purely combinatorial proof of the Vaught conjecture.

In a paper several years ago, Faith [2] proved an extension of a well-known theorem of Kaplansky [4]. His proof, even for the division ring case, was somewhat complicated. Using an old trick of Brauer [1] we show how Faith's theorem follows from Kaplansky's immediately.

In this note we consider the class of topological spaces whose every Hausdorff quotient is regular (or equivalently, normal). Sufficient conditions in order that a space belongs to this class have appeared in papers by Morita [3], and more recently by MacDonald and Willard [2]. In both cases, the authors impose conditions implying paracompactness. We shall derive additional sufficient conditions here which show that paracompactness is not necessary in general.