Define
$$(\Delta w_j, \ \Delta w_k)=\iiint \Delta w_j \cdot \Delta w_k=(\Delta w_k, \ \Delta w_j)$$
Then
\begin{align} \left( \sum c_k \Delta w_k, \ \sum c_j \Delta w_j \right) &= (c_1 \Delta w_1 + \dots + c_n \Delta w_n, \ c_1 \Delta w_1 + \dots + c_n \Delta w_n) \\\\ &= \iiint c_1^2 \Delta w_1 \cdot \Delta w_1 + \dots + c_n^2 \Delta w_n \cdot \Delta w_n + c_1 c_2 \Delta w_1 \cdot \Delta w_2 \\ & \ \ \ \ + c_2 c_1 \Delta w_2 \cdot \Delta w_1 + \dots d \mathbf{x} \\\\ &= \sum (\Delta w_j, \Delta w_j) c_j^2 + 2 \sum \sum_{j \neq k} (\Delta w_k, \ \Delta w_j) c_k c_j \end{align}
Question
Why do the terms $c_1c_2 \Delta w_1 \cdot \Delta w_2 + c_2 c_1 \Delta w_2 \cdot \Delta w_1 + ...$ were added in the triple integral if it's supossed to be a dot product, i.e. $x\cdot y=x_1y_1+\dots+x_ny_n$?