1. Let G be a compact metric 0-dimensional Abelian group. Its dual or character group Γ is a discrete countable torsion group. We denote elements of G by x, of Γ by y, the value of the character y at x by (x, y), and the Fourier transform of f by

The author provides a new and very simple proof of the following theorem of Magnus (2):

Theorem. Let the group G have both a presentation with n + r generators and r relations also a presentation with n generators. Then G is the free group on n generators.

A non-enumerable set of real numbers, Z, is concentrated provided that there exists an enumerable set K such that if G is open and G ⊂ K then Z − G is at most enumerable. Z is said to be universally concentrated if we can take for K any enumerable set H for which

In (1), Theorem 4·1·4, and hence Theorem 4·3·1, are incorrect as stated, unless h* denotes ordinary cohomology theory. However, if h* denotes complex K-theory, corresponding results still hold, provided everything is graded over Z2 instead of Z.

The Peter-Weyl theorem states that for a compact topological group G, the set of finite dimensional (almost invariant) functions on G is uniformly dense in the set of continuous functions on G. In this paper, we consider the question: which subalgebras of C(G) inherit this property?

In a recent paper (1) Effros defined a topology for the hyperspace of nonempty closed sets of a topological space X, and showed that, if X is polonais, the Borel structure of is standard. We shall show that this conclusion will also follow if X is σ-compact and metrizable.

1. A Banach space E over the complex field C is said to be a Banach inner-product (BIP) space if there exists a mapping 〈.,.〉 of E × E into C satisfying:

(i) 〈x, x〉 ≥ 0 (x ∈ E) with equality only if x = 0;

(ii) 〈x, y〉 = 〈y, x〉 (x, y ∈ E);

(iii) 〈x + λy, z〉 = 〈x, z〉 + λ〈y, z〉 (x, y, z ∈ E, λ ∈ C);

(iv) 〈x, x〉 ≤ k2 ‖x‖2 (x ∈ E),

where k is a fixed positive number. Thus 〈.,.〉 is an inner product on E, which induces a norm ‖·‖1 by the relation

This paper is devoted to results concerning the connexion between the ‘abstract’ preK spaces of (l) and the ‘concrete’ Köthe spaces of (3). These results, though in part independent, are not truly original, being in particular known to the authors of (2); but as far as I am aware they have not appeared in print in quite the form they take here.

In this paper I shall be considering the space of linear maps between two (perfect) Riesz spaces, and shall show how certain topological properties of these maps are related to the natural order structure of the space. The fundamental result is (e) of section 4, certain special cases of which have been treated in (4) and (5), using a less elliptic method of proof. Probably the most interesting new result in the present paper is section 10 in the special case of both L and M× being L1 spaces (so that |σ| (L, L×) and |σ| (M×, M) are the norm topologies ((2 b), section 7) and Λ(L; M×) is the space of norm-continuous linear maps).

A K-space (E, τ) is a linear space E of sequences with a locally convex topology τ for which the inclusion map: (E, τ) → (ω, product topology) is continuous. In (2) topological properties of K-spaces were determined directly from properties of the space E and the topology τ. It is, however, very natural to consider duality properties of K-spaces and the purpose of this paper is to determine some of these properties.

In applying the theory of infinite Markov chains to practical examples, it is important to know how the ergodic properties defined by the infinite stochastic or substochastic matrix under consideration are related to those of the n × n (n = 1, 2, 3, …) truncated corner sub-matrices. In particular, it is of interest whether the relevant eigenvalues and eigenvectors of the truncated matrices in some sense approximate to corresponding quantities for the infinite matrix as n → ∞.

A version of Corollary 2 for Hilbert space was orally communicated to the author by C. Pearcy. His proof employed facts about the numerical range, inner products, and orthogonality, and thus made use of structure peculiar to Hilbert space. A Banach algebra with identity is a more comfortable setting for these results, and there it is much clearer just what is important in each proof.

Theorem. if a ≥ 2, b ≥ 1, and a + b ≥ ℵ0, then every collection of more than ab sets each of cardinal not exceeding b contains a subcollection of more than ab sets every two of which have the same intersection.

We denote by ω the linear space of all sequences of real or complex numbers. A linear subspace of ω is called a sequence space. A subset A of ω is solid if whenever x ∈ A and |yi| ≤ |xi| for each i, then y ∈ A. The theory of solid sequence spaces, topologized in a variety of ways, has been developed in considerable detail, in particular by Köthe and Toeplitz (13) and subsequently by Köthe (see, for example (12)). These results have been generalized to function spaces by Dieudonné(6), to vector-valued sequence spaces by Pietsch (18), to vector spaces with a Boolean algebra of projections by Cooper ((4), (5)), and in the real case, to partially-ordered spaces by Luxemburg and Zaanen (see, for example (14)) and Fremlin (8). This last generalization shows that many of the properties of solid sequence spaces depend upon their order structure, rather than upon their structure as sequence spaces.

In a recent paper (5) the author gave an alternative proof of the result due to Hille(2) for the representation of the infinitesimal generator of the Poisson operator. In this paper we obtain the n-dimensional analogue of the result by using arguments similar to those employed in (5).

In a recent note(1) I proved that if μ(n) denotes the usual Möbius function, N denotes a fixed positive integer and if

then

where T runs through all natural numbers ≤ x which are not divisible by an Nth power. In the present paper I shall establish some further relations of this character and, in particular, I shall prove that if

where

then

Thus, in some respects, L(x) appears more regular than M(x), the sum over L(x/T) being multiplicative, whereas M(x1/N) is not.

On Uniform Interpolation Sets

By J. P. Earl

University of Kent

In the paper referred to in the title (these Proceedings 62 (1966), 721–742) the value of a constant used in the statement of several of the theorems is incorrect. Due to the erroneous appearance of a factor ¼ρ2 in the proof of Theorem 3 (penultimate equation of section 5), the value, or range of values, of τ should in fact be 4/ρ2 times the stated value.

There are also some misprints which escaped earlier detection. In the reference of Theorem A d = 0 should read d = D and in the second line of Proof of Theorem 3 Un should be defined by Un ≡ |z − zn| ≤ Δ |zn|1−½ρ. The right-hand side of equation (5.2) should be multiplied by σM, N(z).

Let λ > 0 and let

If , we say that the series is summable (P, λ) to s.

In this paper we give a new generating function for the Jacobi polynomials. The result obtained is of general character and includes as particular cases some of the results given earlier by Carlitz(2), Salam(1), Manocha and Sharma (3,4).

The main aim of the present paper is to give a new generating function for a product of n-Jacobi polynomials. The result established in this paper is the extension of my formula (4).