Math and science::Analysis::Tao::07. Series

Series laws Ⅱ: sums over finite sets

9 basic properties of sums over finite sets. The proofs of these are a good exercise to refresh the notions of functions, bijections, sets, sums and induction.

There is 1-1 correspondence between many of these properties and the 6 properties already shown for sums of the form \( \sum_{i=1}^{n}a_i \). Thus, proving many of them amounts to choosing a bijection that allows transitioning to one of the original properties.

Question: when reading the below properties, consider:

what is the equivalent \( \sum_{i=1}^{n}a_i \) form, if it exists?
does the property extend to sums over infinite sets?

1. Empty set. If \( X \) is empty, and \( f: X \rightarrow \mathbb{R} \) is a function (i.e., \( f \) is the empty function), we have

\[ \sum_{x \in X} f(x) = 0 \]

2. Single element set. If \( X \) consists of a single element, \( X = \{x_0\} \), and \( f: X \rightarrow \mathbb{R} \) is a function, we have

\[ \sum_{x \in X}f(x) = f(x_0) \]

3. Substitution, part I. If \( X \) is a finite set, \( f: X \rightarrow \mathbb{R} \) is a function, and \( g: Y \rightarrow X \) is a bijection, then

\[ \sum_{x \in X}f(x) = \sum_{y \in Y}f(g(y)) \]

4. Substitution, part II. Let \( n \le m \) be integers, and let \( X \) be the set \( X := \{i \in \mathbb{Z} : n \le i \le m\} \). If \( a_i \) is a real number assigned to each integer \( i \in X \), then we have

\[ \sum_{i=n}^{m}a_i = \sum_{i \in X}a_i \]

5. Disjoint sets. Let \( X \), \( Y \) be disjoint finite sets (\( X \cap Y = \emptyset \)), and \( f: X \cup Y \rightarrow \mathbb{R} \) is a function. Then we have

\[ \sum_{z \in X \cup Y}f(z) = \left( \sum_{x \in X} f(x) \right) + \left( \sum_{y \in Y} f(y) \right) \]

6. Linearity, part I. Let \( X \) be a finite set, and let \( f: X \rightarrow \mathbb{R} \) and \( g: X \rightarrow \mathbb{R} \) be functions. Then we have

\[ \sum_{x \in X}(f(x) + g(x)) = \sum_{x \in X}f(x) + \sum_{x \in X}g(x) \]

7. Linearity, part II. Let \( X \) be a finite set, and let \( f: X \rightarrow \mathbb{R} \) be a function, and let \( c \) be a real number. Then

\[ \sum_{x \in X}cf(x) = c\sum_{x \in X}f(x) \]

8. Monotonicity. Let \( X \) be a finite set, and let \( f: X \rightarrow \mathbb{R} \) and \( g: X \rightarrow \mathbb{R} \) be functions such that \( f(x) \le g(x) \text{ for all } x \in X \). Then

\[ \sum_{x \in X}f(x) \le \sum_{x \in X}g(x) \]

9. Triangle inequality. Let \( X \) be a finite set, and let \( f: X \rightarrow \mathbb{R} \) be a function, then

\[ |\sum_{x \in X}f(x)| \le \sum_{x \in X}|f(x)| \]

The 4 sets of series laws: part Ⅱ

Unique to finite sets: The first two properties are trivial and don't appear elsewhere.
Applies to finite sets but not infinite sets (converging series): Substitution 1 doesn't apply to infinite sets as convergence can't be guaranteed (just imagine the function \( g(i) = 1 \). Substitution 2 is inherent in the definition of infinite set sums, so does not appear as a law. The triangle inequality doesn't apply to infinite sets as their sum is only defined for absolute convergence. I'm not sure why monotonicity isn't defined for infinite sets. converging series, there is no triangle inequality defined as there is no definition (thus no value) for their conditional convergence. Tao doesn't absolutely define a comparison test for absolutely converging series either but I'm not sure why (maybe because it is covered by infinite series?)
Equivalent between finite set and finite series notation: The 2nd substitution rule translates between the two forms, so in a way, part I and part II laws have equivalency.

Context

finite series → finite sets → infinite series → infinite sets (absolutely convergent series)