A name for the property $ \| x \star y \| = \| x \| \| y \| $. Suppose that $ \star: V^2 \to V $ is some binary operation on a vector space $ V $. Should it hold, is there a name for the following property?
$$
\forall x,y \in V: \quad
\| x \star y \| = \| x \| \| y \|.
$$
 A: Let $ \Bbb{F} $ denote the base field of $ V $. If we assume that $ \star $ is a binary operation on $ V $ that turns $ V $ into an $ \Bbb{F} $-algebra, i.e.,


*

*Left distributivity: $ x \star (y + z) = x \star y + x \star z $ for every $ x,y,z \in V $,

*Right distributivity: $ (x + y) \star z = x \star z + y \star z $ for every $ x,y,z \in V $, and

*Compatibility with scalar multiplication: $ \lambda \cdot (x \star y) = (\lambda \cdot x) \star y = x \star (\lambda \cdot y) $ for every $ \lambda \in \Bbb{F} $ and every $ x,y \in V $,


then we call $ \| \cdot \| $ a multiplicative norm for $ \star $.
Note: As we are discussing norms here, we necessarily assume that $ \Bbb{F} \in \{ \Bbb{R},\Bbb{C} \} $.
It turns out that if $ V $ is a unital $ \Bbb{R} $-algebra having a multiplicative norm, then by a result of Urbanik and Wright, $ V $ is isomorphic to one of the following four normed $ \Bbb{R} $-algebras:


*

*$ \Bbb{R} $ (the real field).

*$ \Bbb{C} $ (the complex field).

*$ \Bbb{H} $ (the quaternions).

*$ \Bbb{O} $ (the octonions).

