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Formal Sum in Mathematics 📂Lemmas

Formal Sum in Mathematics

Definition

For a set $S = \left\{ s_{1}, s_{2}, \cdots , s_{n} \right\}$ and a field $\mathbb{F}$, the following notation is called a formal sum.

$$ \sum_{i} a_{i} s_{i} = a_{1} s_{1} + a_{2} s_{2} + \cdots + a_{n} s_{n}, \qquad \text{where } a_{i} \in \mathbb{F} $$

  • $S$ need not be a finite set.
  • $\mathbb{F}$ need not actually be a field.

Explanation

Note that the set $S$ is merely a set and does not carry any algebraic structure. In particular, the addition symbol $+$ appearing in the above definition is not a binary operation. The expression $a_{1}s_{i} + \cdots + a_{n}s_{n}$ is itself a single object determined by $a_{i}$ and $s_{i}$. In other words, it is essentially like $2n$ ordered pairs of $a_{i}$ and $s_{i}$.

$$ a_{1}s_{i} + \cdots + a_{n}s_{n} \equiv (a_{1},s_{1},a_{2},s_{2},\cdots,a_{n},s_{n}) $$

One advantage of using the addition notation is that it allows a concise expression using the symbol $\sum$.

$$ \sum_{i} a_{i}s_{i} = (a_{1},s_{1},a_{2},s_{2},\cdots,a_{n},s_{n}) $$

Also, viewing this in the context of a vector space, the set of formal sums of elements of $S$ can be simply written as follows.

$$ \operatorname{span} (S) = \left\{ \sum_{i} a_{i}s_{i} : a_{i} \in \mathbb{F}, s_{i} \in S \right\} $$

One may also denote these by $\mathbb{F}S$ or $\mathbb{F}[S]$. From the above expression, $S$ can be regarded as a basis, and $\mathbb{F}[S]$ can be seen as the formal vector space generated by $S$. It is called “formal” because no substantive addition is defined between the vectors inside $\mathbb{F}[S]$.

See also