Definition talk:Dot Product/Real Euclidean Space

This definition for the dot product should be restricted to only Euclidean space. In a generic normed vector space, it not only fails to be equivalent to Definition 1, but it fails to be an inner product.

First, observe:

Now, consider $\R^2$ with the taxicab norm:

But:

This violates Axiom $2$.

I was considering suggesting that the definition of angle between vectors be modified as well, but there may be an interesting geometric interpretation of that, so I'll leave it be. --CircuitCraft (talk) 19:10, 27 April 2023 (UTC)


 * Clearly then the two definitions cannot be considered as equivalent, but that definition $1$ is more general than $2$ which is a special case of $1$.


 * All too much for me at the moment. I think I need to bow out. --prime mover (talk) 22:10, 27 April 2023 (UTC)