In real Chebyshev approximation by generalized rational functions, constraining denominators to be positive guarantees that the set of best coefficients is convex [1, 181]. We show by means of an example that denominators must be constrained to be of constant argument for such a convexity result to hold in complex Chebyshev approximation.

J. Dixmier has observed [3, p. 1070] that a non-reflexive Banach space has non-rotund fourth conjugate space. It is the aim of this paper to improve Dixmier’s result by showing that a non-reflexive Banach space already has non-smooth third conjugate space in that the images under natural embedding of the continuous linear functionals which do not attain their norm on the unit sphere are non-smooth points of the third conjugate space.

Let C(n) be the number of partitions of n such that all even multiplicities of the parts are less than 2m, m>1; and all odd multiplicities are at least (2r+1) and at most 2(m+r)—1, r≥0. Let D(n) be the number of partitions ofn into parts which are either odd multiples of (2r+1) or are even and not divisible by 2m.

Let A be a commutative algebra contained in Mn(F), F a field. Then A is nilpotent if there exists v such that Av=(0), and is said to have nilpotency class k (denoted Cl(A)=k) if Ak=(0), but Ak-1≠(0). A well known result asserts that matrix algebras are nilpotent if and only if every element is nilpotent. Let N = {A | A is a nilpotent commutative subalgebra of Mn(F)}.

In this note we give a simple proof of an operator-interpolation theorem (Theorem 2) due originally to Donoghue [6], and Lions-Foias [7].

Let be the complex plane, the open upper half-plane, the real line, ℛ+ and ℛ- the non-negative and non-positive axes. Denote by the class of positive functions on which extend analytically to —ℛ-, and map into itself. Denote by ’ the class of functions φ such that φ(x1/2)2 is in .

Let G be a group, ZG its integral group ring and U(ZG) the group of units of ZG. The elements ±g∈U(ZG), g∈G, are called the trivial units of ZG. In this note we will prove

Let G be a finite group. If ZG contains a non-trivial unit of finite order then it contains infinitely many non-trivial units of finite order.

In [1] S. D. Berman has shown that if G is finite then every unit of finite order in ZG is trivial if and only if G is abelian or G is the direct product of a quaternion group of order 8 and an elementary abelian 2-group.

Let A and B be positive semidefinite Hermitian n-square matrices. If A—B is positive semidefinite, write A≥B. Haynsworth [1] has proved that if A≥B then det(A+B)≥det A+n det B.

Let G be a subgroup of the symmetric group, Sn, and let λ be a character on G. Let

where A = (aij) and Er is the rth elementary symmetric function.

In [1] Lambek calls the injective R-module I nice if every torsionfree factor module of the ring of quotients Q of R with respect to lis divisible. If lis nice then g is a dense subring of the bicommutator BicRI of I with respect to the finite topology (see [1, Proposition 2]). We now give an example of an injective R-module over an Artinian ring R which is not nice. Since R is Artinian, Q=BicRI, by Proposition B of [1].

Before we give the example, we state the following, which depends on [2] for terminology.

Purdy asked whether the following conjecture is true

Conjecture. Let E be a set of 2n elements. If S={Sl, S2, …, St} is a Sperner system of E, i.e. for i≠j, i, j, =1, 2, …, t; and if

(1)

then

The proof of the conjecture will be obtained using the following theorem of Katona (Acta Math. 15 (1964), 329-337):

Let N be a left near ring and let M be a right N-module. We recall [1] that M is called injective iff every diagram can be embedded into a commutative diagram where A and B are right N-modules with exact.

The purpose of this note is to show that if N is a d.g. near-ring with identity, then M is injective iff for every right ideal u of N and every N-homomorphism f:u→N, there exists an element m in M such that f(a)=ma for all a in u.

Let A be a uniform algebra on the compact set X and let ψ be a non-trivial linear functional on A. A finite non-negative measure μ on X is called a Jensen measure for ψ if

(1)

Let T be a linear operator on a Banach space X and consider the sequence of ranges

where the inclusions are not necessarily proper. The linear subspaces Xn=TnX (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X0=X and denote by |x|0 (x∈X0) the norm of X0.