Limit Lim Research Articles

On mackey convergence in locally convex spaces

After some introductory propositions, we give a dual characterization of those locally convex spaces which satisfy the Mackey convergence condition or the fast convergence condition by means of Schwartz topologies. Making use of the universal Schwartz space (l∞,τ(l∞,l1)) we prove some representation theorems for bornological and ultrabornological spaces, that is, every bornological spaceE is a dense subspace of an inductive limit lim indEa, a∈A, ofseparable Banach spacesEa, and every Mackey null sequence inE is a null sequence in someEa. IfE is ultrabornological, thenE can be represented as lim indEa,a∈A, allEa separable Banach spaces, such that every fast null sequence inE is a null sequence in someEa.

On the boundary behavior of the derivative of analytic functions

In 1929 R. Nevanllnna [5] introduced the class of functions/(z), analytic and hounded in the unit circle ] z l < l , for which the radial limits lim /(re ~x) are of r -~ l -0 modulus 1 for almost all x in the interval 0 ~< x ~< 2 ~. This class will here be called class (N). The set of arguments x, such tha t lira /(re t~) does not exist r ~ 1 0 or is not of modulus 1, will be called the exceptional set of the function [(z). Each function in class (N) admits a representation

On the probability that n and g(n) are relatively prime

o x o oga) and satisfies condition (B) of 5 2, then the probability in question exists and is equal to 6~~~. Condition (B) means roughly that f(x) increases faster than the function logil:log,a. In $ 3 we show that condition (B) is the best possible. Condition (A) may be perhaps relaxed; but it, cannot be replaced by f(a) = O(x/logloglog~). We also consider the average number of divisors of (,N, g(,n)). This is the limit lim[H(X)/z}, where S(x) is t,he sum of the numbers of divisors of all nuZYers (98, g(w)), n < x. We assume throughout8 that f(m) is a monotone increasing positive function with a piecewise oontinuous derivative; P(y) will denote the inverse of f(x). By v, ,u, C, c7 we denote the standard number-theoretic functions, by log,s, log,m, . . . the iterated logarithms of 8. We begin with some elementary identities. Let Qk($) be the number of integers ti ,< x: such that n and g(n) have no common factors < k. If H(m, d) is the number of IZ < x with dl(n, g(n)), then

On the continuity and limiting values of functions

of the limting values of / as the independent variable £ approaches x and we can also define the continuity of / at some of the points xdX. These topological structures on X will be such that for a noncountable set of points xdX the limiting values of / as £—>x and the continuity of / at x can be interpreted in at least two different ways. For example if X is the real line one an speak about the limiting values of / as £ approaches x from the left and from the right, and of limj^.I_o/(£) and lim{_I+o/(£), and of the left and right continuity of / at x. The exact definition of these topological structures on X is given later. We shall prove two general theorems for such functions: One of these concern the equality of the limits lim/(£) as £ approaches the same point £ under different conditions. Two cases can be distinguished according as lim/(£) is uniquely determined for each of these topological structures or not. The other theorem states that except for a negligible subset of X the function / is continuous with respect to both or neither of the topologies defined on X. These theorems include many old and new results as special cases from the theory of functions of one and of several real variables. For better understanding it is perhaps more suitable to discuss first some of the known results and to state the new ones afterwards. For real valued functions of one real variable some precise results of this type have been known for a long time. The common source of one group of these results is the following theorem due