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- Jun 22, 2012

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\(\displaystyle \mathbb{R}^n\) is a vector space but not a field because it lacks a suitable multiplication operation between pairs of its elements ...

Why don't mathematicians define a multiplication operation between a pair of elements and investigate the resulting field ...

For example ... why not define multiplication as X where

\(\displaystyle (x_1, x_2, \ ... \ ... \ , x_n) \ X \ (y_1, y_2, \ ... \ ... \ , y_n) = (x_1 y_1, x_2 y_2 , \ ... \ ... \ , x_n y_n)\) ...

Peter

Why don't mathematicians define a multiplication operation between a pair of elements and investigate the resulting field ...

For example ... why not define multiplication as X where

\(\displaystyle (x_1, x_2, \ ... \ ... \ , x_n) \ X \ (y_1, y_2, \ ... \ ... \ , y_n) = (x_1 y_1, x_2 y_2 , \ ... \ ... \ , x_n y_n)\) ...

Peter

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