In this note we show that a paranormal operator T on a Banach space satisfies Weyl's theorem. This is accomplished by showing that

(i) every isolated point of its spectrum is an eigenvalue and the corresponding eigenspace has invariant complement,

(ii) for α ≠ 0, Ker(T-α) ⊥ Ker (T-β) (in the sense of Birkhoff) whenever β ≠ α.

It is shown that, for a semi-topological functor T: A → X, the category A is total, that is, the Yoneda embedding of A has a left adjoint, if X is total. In particular, monadic categories over Set (possibly without rank) are total, and full reflective subcategories of total categories are total.

Integral comparison theorems of Hille-Wintner type of second order linear equations are shown to be valid for the third order linear equation y‴ + q(t)y = 0.

Spaces with homogeneous norms are closely related to the Beppo Levi spaces of Deny and Lions, to spaces of Riesz potentials, and to Sobolev spaces. In this paper we survey the literature on them, give a broad extension of their definition, and present their basic theory. Many of the properties of Sobolev spaces have their analogues. In fact, the two families are locally equivalent. Spaces with homogeneous norms are especially suited to the study of boundary value problems on for homogeneous elliptic operators with constant coefficients. We will use them extensively in a forthcoming paper to study elliptic partial differential equations with mixed boundary conditions on a smoothly bounded domain.

We discuss a computationally stable numerical method for the solution of linear programs and games. The method is useful in obtaining approximate solutions to large numerically unstable linear programs.

The main theorem of this paper shows that a complex p-normed algebra which is a pre-Bezout domain is isomorphic to the field of complex numbers, if it is a generalized unique factorization domain. This theorem generalizes the previous result of the authors proved by them in their paper Bull. Austral. Math. Soc. 20 (1979), 247–252. Some applications are then given.

Criteria for a holomorphic function f defined in |z| < 1 to be Bloch, namely, , are obtained in terms of the area and the length of the images of non-Euclidean disks and non-Euclidean circles, respectively.

It was shown by Bryce and Cossey that each Hall π-subgroup of a group in the smallest normal Fitting class S* necessarily lies in S*, for each set of primes π. We prove here that for each set of primes π such that |π| ≥ 2 and π′ is not empty, there exists a normal Fitting class without this closure property. A characterisation is obtained of all normal Fitting classes which do have this property.

Let F be a normal Fitting class closed under taking Hall π-subgroups, in the sense of the paragraph above, and let Sπ denote the Fitting class of all finite soluble π-groups, for some set of primes π. The second main theorem is a characterisation of the groups in the smallest Fitting class containing F and Sπ in terms of their Hall π-subgroups.

Singular inner functions are in many respects the most important and difficult type of functions in the Banach algebra H∞ of bounded analytic functions in the unit disc. This paper is concerned with ideals generated by singular inner functions. In particular, conditions on the associated measures are given so that the ideal spans the whole algebra H∞. To this end the local boundary behavior of a singular inner function is studied and the results obtained there may be of independent interest.

The aim of this paper is to study the relationship between the nature of an infinite series of real terms in which the general term tends to zero and the derived set (set of limit points) of the aggregate of the fractional parts of its partial sums. For all types of series (in which the nth term tends to zero) we determine the derived set.

We denote by F the set of the fractional parts of the partial sums of the series, and by F′ the derived set of F. The principal results of the paper can be stated as follows:

1. the series is convergent if and only if F has at most one limit point in [0, 1] or F′ ⊆ {0, 1};

2. the series, if non-oscillatory, is divergent if and only if F′ = [0, 1];

3. if the series is oscillatory, F′ is a closed sub-interval of [0, 1] or [0, 1] - (a, b), where (a, b) ⊊ (0, 1).

We prove a Baire category theorem for continuous lattices and derive category theorems for non-Hausdorff spaces which imply a category theorem of Isbell's and have applications to the spectral theory of C*-algebras. The same lattice theoretical methods yield a proof of de Groot's category theorem for regular subcompact spaces.

We are interested in the strict topology τ on , the set L(E, F) of all continuous linear mappings from E into a Banach space F endowed with the topology of pointwise convergence. The T3-completion of the convergence tensor product E ⊗cLc F is the set of all τ-continuous linear functionals on L(E, F) and τ is the topology of uniform convergence on the compact subsets of . In the case that E is a nuclear Fréchet space, a nuclear (DF)-space or a Banach space with the bounded approximation property the topology τ agrees with the topology of Lco (E, F).