I am studying linear algebra using Axler's 3rd edition book.
When checking whether a set is a vector space, I refer to the definition on page 12. These are the definitions being used:
A vector space is a set V along with an addition on V and a scalar multiplication on V such that the following properties hold: commutativity, associativity, additive identity, additive inverse, multiplicative identity, distributive properties.
The book defines the operator "+" to be closed in set V and scalar multiplication to be closed in set V, therefore when I check whether a space is a "vector space" using these two operations, I only check whether the bolded properties hold.
Question 1: It appears that "+" and scalar multiplication are inhereited from the field which the vector space is over. Is this interpretation correct?
Question 2: Suppose now that there is an alternative definition of addition (call it +') and scalar multiplication (call it $\cdot'$) over a vector space candidate W, that does not correspond with our commonly known addition and multiplication in $\mathbb{R}$. To check whether W is a vector space, I know I have to go ahead and check the 7 properties, but I'm not sure whether checking that addition and scalar multiplication are closed is a) redundant or b) necessary. Put in a different way, does a set U following the 7 properties of a vector space imply "closed under addition and scalar multiplication?