An Orlicz-Pettis theorem for conditionally convergent series

Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series

Let X,Y be locally convex spaces and L(X,Y ) the space of continuous linear operators from X into Y . We consider 2 types of multiplier convergent theorems for a series P Tk in L(X,Y ). First, if λ is a scalar sequence space, we say that the series P Tk is λ multiplier convergent for a locally convex topology τ on L(X,Y ) if the series P tkTk is τ convergent for every t = {tk} ∈ λ. We establish...

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map B : X×Y → Z, a sequence x = (xn)n ⊆ X is called B-absolutely summable if ∑∞ n=1 ‖B(xn, y)‖Z < ∞ for any y ∈ Y . Connections of this space with `weak(X) are presented. A sequence x = (xn)n ⊆ X is called B-unconditionally summable if ∑∞ n=1 |〈B(xn, y), z∗〉| < ∞ Preprint submitted to Elsevier 21 December 2007 for any y ∈ Y and z∗ ∈ Z∗...

On rearrangements of conditionally convergent series

Rearrangement of a Conditionally Convergent Series

One of the surprising results in an elementary calculus course is that a rearrangement of a conditionaly convergent series may change its sum, even its very convergence. Observing typical textbook examples of this phenomenon, it turns out that during the rearrangement some of the terms are moved arbitrarily large distances from their original locations. Is this necessary? The answer is positive...

A Distributional Approach to Conditionally Convergent Series

Whether the car’s gas tank is filled up on Monday and the paycheck is deposited on Tuesday, or vice versa, the contribution of those two transactions to the checkbook’s final balance is the same. By the commutative property, order does not matter for the algebraic addition of a finite number of terms. However, for a super banker who conducts an infinite number of transactions, order may matter....