From my understanding, a vector space is a set that is closed under addition and multiplication, so let $A$ denote a set, a vector space $V = (A, +, \times)$
But whenever you read the definition of vector space, it always says something like: "Suppose $V$ is a vector space over the field of ...."
But I thought a vector space is defined in terms of sets, not fields!
Can someone help me resolve this confusion?
On
Vector spaces are not sets closed under addition and multiplication. Those are sets $A$ which are coupled with a field $F$, and with a binary operator called addition $+$, and with scalar multiplication when the scalar is from $F$. And we require that these operations satisfy certain properties.
In general there is no "internal" multiplication on vector spaces. You can only multiply something by a scalar which sits "outside" the vector space.
No, a field $F$ has addition and multiplication between members of $F$ (and has a $0$ and a $1$, etc. plus all the axioms).
A vector space is a set $V$ with addition of members of $V$ (which is an Abelian group, so has a $0$ etc.), and there is also a field $F$ (the field over which we have the vector space) and an extra multiplication called scalar multiplication (scalars are members of $F$) defined as a map $F \times V \rightarrow V$ that satisfies some axioms.
If $F$ is a field itself, we can use $F = V$ and see $F$ as a vector space over itself, using standard multiplication as scalar multiplication. So we can see $\mathbb{R}$ as a vector space over $\mathbb{R}$, but also over $\mathbb{Q}$ (much more interesting!). But in general, $V$ (the vectors) is not a field (only addition is present), but there is a scalar field $F$ (that has to be specified as well) with an extra "cross-domain" multiplication. For the plane we can multiply a point by a number, stretching each coordinate ($a(x,y) = (ax,ay)$, which defines multiplication of $a \in \mathbb{R}$ with $(x,y) \in \mathbb{R}^2$), which is a standard example. The whole of linear algebra is about vector spaces.
In more formal terms: a vector space $V$ over the field $F$ (with its own addition and multiplication $+$ and $\ast$ and its own 0,1) is a tuple $(V,F,+,\times,0)$ where $+ : V \times V \rightarrow V$, $\times: F \times V \rightarrow V$ and $(V,+,0)$ is an Abelian group, and we have the additional axioms (universal quantifiers implied): $a \times (x + y) = a\times x + a \times y$, $(a+b) \times x = a \times x + b \times x$, $1 \times x = x$, $(a \ast b) \times x = a \times (b \times x)$.