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Cauchy-Schwarz Inequality for Finite Sums in Ordered Commutative Semirings

In any linearly ordered commutative semiring $R$ equipped with an additive structure compatible with its order and satisfying the strict ordered ring properties, for any finite set $s$ and functions $f, g : \iota \to R$, the square of the sum of the products $f(i)g(i)$ over $s$ is less than or equal to the product of the sums of the squares of $f(i)$ and $g(i)$ over $s$. Formally,

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Cauchy-Schwarz Inequality for Finite Sums

For any finite set $s$ of indices $\iota$ and functions $f, g : \iota \to \mathbb{R}$, the inequality $\sum_{i \in s} f(i) \cdot g(i) \leq \sqrt{\sum_{i \in s} [f(i)]^2} \cdot \sqrt{\sum_{i \in s} [g(i)]^2}$ holds.

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Cauchy-Schwarz Inequality for Sums in Ordered Commutative Semirings

For a finite set $s$ and non-negative functions $f,g$ over $s$ in an ordered commutative semiring $R$, if $r_i^2 = f_i g_i$ for all $i \in s$, then $\left(\sum_{i \in s} r_i\right)^2 \leq \left(\sum_{i \in s} f_i\right) \cdot \left(\sum_{i \in s} g_i\right)$.

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Cauchy-Schwarz Inequality for Non-negative Real Numbers

For any finite set $s$ of indices $\iota$ and functions $f, g : \iota \to \mathbb{R}_{\geq 0}$, the inequality $\sum_{i \in s} f(i)g(i) \leq \sqrt{\sum_{i \in s} f(i)^2} \cdot \sqrt{\sum_{i \in s} g(i)^2}$ holds, where the square roots denote the principal non-negative square roots defined on $\mathbb{R}_{\geq 0}$.

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Cauchy-Schwarz Inequality for Sums of Square Roots of Non-Negative Functions

For any finite set $s$ and non-negative real-valued functions $f$ and $g$ defined on $s$, the inequality $\sum_{i \in s} \sqrt{f(i)} \cdot \sqrt{g(i)} \leq \sqrt{\sum_{i \in s} f(i)} \cdot \sqrt{\sum_{i \in s} g(i)}$ holds.