How is the notation of "the" dual vector space $V^*$ justified?

Question

My naive idea of an expression is that it returns (or denotes) exactly one object (be that may a class or a set), but of course there are multiple vector spaces being isomorphic to one another, thus there are multiple vector spaces being the dual of $V$.

The question is what do we have to say when studying vector spaces to erase ambiguity?

Should I just use the predicate "$W$ being a dual vector space of $V$", or just scratch a "here we only care about the vector space structures" on the margin?

Edit: I realize now how stupid the question is, what I have in my mind is how when we prove theorems concerning only the vector space structure, the result of that theorem can safely be applied to isomorphic vector spaces, thus isomorphism behaving as a kind of equality.

Answer 1

The dual space of a $k$-vector field $V$ is defined as the vector space $$V^* := \hom(V,k).$$ Here, $\hom$ means the space of linear functions (from the first argument to the second one).

Answer 2

There is just one dual space of $V$: the space of linear functionals. We name that one $V^*$. There are other spaces isomorphic to $V^*$. If you need one of those other spaces $X$ in a proof you show/say that $X$ is isomorphic to $V^*$.

Answer 3

The dual space is the vector space of all linear functions $V\to K$ where $K$ is the field of scalars of $V$ (note that $K$ is part of the definition of the vector space $V$, so there's only one $K$ for the given $V$). There is only one such space. If $W$ is isomorphic to $V$, then $W^*$ is isomorphic to $V^*$, but if $W$ is not $V$, then $W^*$ is not $V^*$.