一〇二上實分析筆記@S-2013.09.09

§1.1 Summability and System of Real Numbers
Let $I$ be index set. For $S$ an arbitrary set, let $\mathcal F(S)$ be the collection of all nonempty finite subsets of $S$

Definition: A System of $\mathbb R$
$\{r_\alpha\}_{\alpha\in I}$ denotes a system of $\mathbb R$, where $r_\alpha\in\mathbb R$.

Definition: Summable
$\{r_\alpha\}_{\alpha\in I}$ is called summable with sum $s$, if for all $\epsilon>0$, there exists an $A\in\mathcal F(I)$ such that $$\left|\sum_{\beta\in B}r_\beta-s\right|<\epsilon$$ whenever $A\subset B\in\mathcal F(I)$.

Theorem 1.1.1
If $\{r_\alpha\}_{\alpha\in I}$, $\{r_\alpha'\}_{\alpha\in I}$ are both summable, and $a,a'$ another two real numbers, then $\{ar_\alpha+a'r_\alpha'\}_{\alpha\in I}$ is summable.

Theorem 1.1.2
If $r_\alpha\geq0$ for all $\alpha$, then $\{r_\alpha\}_{\alpha\in I}$ is summable if and only if $$\sum_{\beta\in B}r_\beta$$ is bounded (above) among all $B\in\mathcal F(I)$.

Theorem 1.1.3
$\{r_\alpha\}_{\alpha\in I}$ is summable if and only if for every $\epsilon>0$, there exists an $A\in\mathcal F(I)$ such that $$\left|\sum_{\beta\in B}r_\beta\right|<\epsilon$$ whenever $B\in\mathcal F(I)$ and $A\cap B=\varnothing$.