An elementary new, simple and purely analytic proof of Laguerre's theorem about the zeros of the polar derivative of a polynomial P(z) with respect to the point α, is given. The proof is based on a lemma which is also of independent interest.

Let J ⊆ C∞ (Rn) be any ideal. Since a function of the variables = (t1,…,tn) is a function of the variables which does not depend on , we have J ⊆ C∞ (IRn+P). Of course, J is not an ideal of C∞ (IRn+P), but it generates an ideal that we call . Consider the following statement (1) on J: “Given any if and only if for every fixed .

In this paper we show that statement (1) holds for a large class of finitely generated ideals although not for all of them. We say that ideals satisfying statement (1) have line determined extensions. We characterize these ideals to be closed ideals J() (in the sense of Whitney) such that for all p ∈ ℕ, the ideal is also closed. Finally, some non-trivial examples are developed.

For a locally compact topological group admitting a weight function, we establish necessary and sufficient criteria for all the weighted continuous functions to be weakly almost periodic. Among other results, we show that weak almost periodicity of all ω-weighed continuous functions on a discrete semigroup S, can be very different drom the phenomenon of regularity of multiplication in the weighted algebra ℓ1 (S, w).

Some properties which different definitions or orthogonality in a normed space can possess are considered. It is shown that orthogonality can be defined on any separable space with many of the properties possessed by the usual orthogonality in an inner-product space, but that the possession of a further property forces the space to be isomorphic to a Euclidean space.

In this paper, we prove the existence of a continuous one-to-one correspondence between bounded solutions of the equation x′(t) = Ax(t) + b(t) which belong to a certain subclass L+ of and bounded mild solutions of the equation x′(t) = Ax(t) + b(t) + f(t, x(t)) of the form u(t) = ø(t) + Ψ(t), with ø(t) ∈ L+ and .

We study a notion of smoothness of a norm on a Banach space X which generalizes the notion of uniform differentiability and is formulated in terms of unicity of Hahn Banach extensions of functionals on block subspaces of a fixed Schauder basis S in X. Variants of this notion have already been used in estimating moduli of convexity in some spaces or in fixed point theory. We show that the notion can also be used in studying the convergence of expansions coefficient of elements of X* along the dual basis S*.

For a homogeneous C*-algebra we identify the quotient of the automorphism group by the locally unitary automorphisms as a subgroup of the homeomorphisms of the spectrum. We sharpen a known criterion on the spectrum that ensures that all locally unitary automorphisms of the algebra are inner.

Let A be a noncommutative C*-algebra other than M2(I). We show that there exists a completely positive map φ of norm one on A and an element a ɛ A such that φ(a) = a, φ(a*a) = a*a, but φ(aa*) ≠ aa*.