Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

In this note we prove the following theorem, which does not seem to appear explicitly in the literature.

Theorem. Let A be a finite abelian group and p the smallest prime which divides |A|. Then there are p—1 mutually orthogonal quasigroups of order |A|, one of which is A.

A well-known embedding theorem of Banach and Mazur [1, p. 185] states that every separable Banach space is isometrically isomorphic to a subspace of C[0, 1], establishing C[0, 1] as a universal separable Banach space. The embedding theorem one encounters in a course in topological vector spaces states that every Hausdorff locally convex space (l.c.s.) is topologically isomorphic to a subspace of a product of Banach spaces.

The following result is proved in [1, p. 6].

Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.

Let E be a Hausdorff topological vector space. A subset A of E is a polytope iff A is the convex hull of a finite number of points. In this note a necessary condition for every maximal convex subset of a subset B of E to be a polytope is given. This is related to a problem first posed by Klee [1] for compact three-cells in Euclidean 3 space.

All spaces in this note are regular, Hausdorff topological spaces.

At the topology conference in Pullman, Washington, in March 1970, E. Michael posed the following question: if X is Lindelôf Y is separable and metrizable and I × Y is paracompact, must X × Y be Lindelöf?