While approaching tensors, I faced what is considered to be the cartesian product of 2 vector spaces. Now as much as I understood what this practically mean, I was quite concerned by its definition.
My concern comes from how the cartesian product is defined, how a vector space is defined and how these definitions seem to collide in the statement" Cartesian Product of Vector Spaces".
The Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B....
A vector space is defined as a quadruple (V,K,⊕,⊙) where V is a set of elements called vectors, K is a field** (K,+,⋅) (and we say that the vector space is a space over K) , ⊕ is a binary operation (called sum) on V such that (V,⊕) is a commutative group and ⊙:K×V→V is a scalar multiplication such that....
Now, a Vector Space is defined as a quadruple while Cartesian Product operates on sets. This is why, even if I understood that for Vector Space is often meant the set V, this created some confusion, at least for me.
I don't know if this naming-statement shortcut/convention is used with other mathematical structures being something common. However as a beginner trying to make sense of all definitions, and trying to understand how they all tie together and interrelate, I was just confused by it.
Are my intuitions right about what it practically mean to say " A Cartesian Product of Vector Spaces"?
Is this a common naming convention? If yes, why is it useful?