Let X, Y be Banach spaces. Their product X × Y with norm
||(x,y)|| = max {||x||, ||y||}
Is also a Banach space.
The natural space of economic models is Rn, the home space of consumption and production sets, which is a typical finite-dimensional normed linear space. In these spaces the interaction between linearity and topology is most acute, and many of the results obtained above can be sharpened. The most important results are summarized in the following proposition