Purchase and price tuple/Duality/Motivation/Example

In a shop, there are ${}n$ different products. A purchase is described by an ${}n$-tuple (purchase tuple), where the ${}i$-th entry says how much of the ${}i$-th product is bought (with respect to a certain unit). The set of all purchases (including refunds) forms an ${}n$-dimensional vector space. A price list for the products is also described by an ${}n$-tuple (price tuple), where the ${}i$-th entry tells the price of the ${}i$-th product (with respect to the same unit). The set of all price tuples forms also an ${}n$-dimensional vector space (think about comparing prices, inflation, tax, etc.). It is obviously nonsense to consider a purchase tuple and a price tuple in the same vector space and to add them. To the contrary, the right way to process a purchase tuple ${}\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)$ and a price tuple ${}\left(p_{1},\,p_{2},\,\ldots ,\,p_{n}\right)$ is to compute the total price ${}\sum _{i=1}^{n}p_{i}x_{i}$, which belongs to ${}\mathbb {R}$. Purchase tuple and price tuple are dual to each other, the vector space of price tuples is dual to the vector space of purchase tuples.