An interesting new metric and its applications to alternating series. (English) Zbl 1023.40002

There are several papers that deal with the application of metric spaces in the theory of series. In these papers often the Fréchet metric \(d_F\) and the Baire metric \(d_B\) are used. In an earlier paper [cf. M. Dindoš, Real Anal. Exch. 25, 599-616 (2000; Zbl 1011.40002)] the author introduced the metric \(d_E\). All these metrics have the unpleasant property that the distance between two series is substantially independent on the rests of the series after the \(n\)-terms. In the present paper the author defines a new metric \(d_D\) without this property.
Let \(X\) be a Banach space with the norm \(|\cdot |\). Let \(M\subseteq X\) be a closed bounded set. If \[ a=(a_n)^\infty_1 \in M^N\quad \text{and} \quad b=(b_n)^\infty_1 \in M^N, \] then we put \[ \begin{split} d_D(a,b)=\\ \sup\left\{ \left|{a_1-b_1\over 1}\right |,\left |{a_1+ a_2-b_1-b_2 \over 2}\right|, \dots,\left |{a_1+ a_2+\cdots+ a_n-b_1- b_2-\cdots- b_n\over n}\right |, \dots\right\}.\end{split} \] It is easy to show that \(d_D\) is a metric and the metric space \((M^N,d_D)\) is complete.
Let \(x_n\in M\) \((n=1,2,\dots)\), and \({\mathfrak A}\) be the set of all \(a\in X\) such that there exist \(n_1<n_2<\dots\) so that \[ a=\overset \bullet {\lim_{k\to\infty}} {x_1+x_2+ \cdots+ x_{n_k}\over n_k}. \] \({\mathfrak A}\) is called the set of averages of \((x_n)^\infty_1\). Here \(\overset \bullet \lim\) denotes the limit in the weak topology of \(X\). If \({\mathfrak A}\) consists of one point then we write \[ A\bigl((x_n) \bigr)=\overset \bullet {\lim_{n\to\infty}} {x_1+x_2+ \cdots+ x_n \over n}. \] Let \(X\) be a Banach space and \(M\) be a closed and bounded set with at least 2 elements. The author proves that for any closed subset \(F\) of \(X\) the set \[ T_F=\biggl \{(x_n)^\infty_1\in M^N: A\bigl((x_n) \bigr)\text{ exists and }A\bigl((x_n)\bigr) \in F\biggr\} \] is a nowhere dense set in the complete metric space \((M^N,d_D)\). The porosity of the set \(T_F\) is also described. Some applications to convergence properties of series of type \[ \sum^\infty_{n=1} (-1)^{a_n}\cdot b_n\tag{1} \] are given, where \(b_n\geq 0\) \((n=1,2,\dots)\), \(\sum^\infty_{n=1} b_n=+\infty\) and \((a_n)^\infty_1 \in\{0,1\}^N.\)
It is proved that if \(\liminf_{n \to\infty} n\cdot b_n>0\), then the set \(C\) of all \((a_n)^\infty_1 \in \{0,1\}^N\) such that series (1) converges, is a nowhere dense set. The porosity of \(C\) is also shown. Similar results hold for the set of all \((a_n)^\infty_1 \in\{0,1\}^N\) such that the sequence of partial sums of (1) is bounded.
These last results extend some earlier results of the author and other authors achieved by applying the metrics \(d_F\), \(d_B\), and \(d_E\).


40A05 Convergence and divergence of series and sequences
26A21 Classification of real functions; Baire classification of sets and functions
54E52 Baire category, Baire spaces


Zbl 1011.40002