ThmDex – An index of mathematical definitions, results, and conjectures.

Result R4038 on D1239: Standard N-operation
Subresult of R3636: Finite product of finite real sums

Product of two finite real sums

Formulation 0

Let $x_1, \ldots, x_N \in \mathbb{R}$ and $y_1, \ldots, y_M \in \mathbb{R}$ each be a D993: Real number.

Then \begin{equation} \left( \sum_{n = 1}^N x_n \right) \left( \sum_{m = 1}^M y_m \right) = \sum_{n = 1}^N \sum_{m = 1}^M x_n y_m \end{equation}

Formulation 1

Let $x_1, \ldots, x_N \in \mathbb{R}$ and $y_1, \ldots, y_M \in \mathbb{R}$ each be a D993: Real number.

Then \begin{equation} \left( x_1 + \cdots + x_N \right) \left( y_1 + \cdots + y_M \right) = (x_1 y_1 + \cdots + x_1 y_M) + \cdots + (x_N y_1 + \cdots + x_N y_M) \end{equation}

Proofs

Proof 0

Let $x_1, \ldots, x_N \in \mathbb{R}$ and $y_1, \ldots, y_M \in \mathbb{R}$ each be a D993: Real number.

By direct computation \begin{equation} \begin{split} \left( \sum_{n = 1}^N x_n \right) \left( \sum_{m = 1}^M y_m \right) & = \left( x_1 + x_2 + \cdots + x_N \right) \left( \sum_{m = 1}^M y_m \right) \\ & = x_1 \sum_{m = 1}^M y_m + x_2 \sum_{m = 1}^M y_m + \cdots + x_N \sum_{m = 1}^M y_m \\ & = \sum_{m = 1}^M x_1 y_m + \sum_{m = 1}^M x_2 y_m + \cdots + \sum_{m = 1}^M x_N y_m \\ & = \sum_{n = 1}^N \sum_{m = 1}^M x_n y_m \end{split} \end{equation} $\square$