Okay first of all here's what I understand of the two notions —
A natural Isomorphism between two sets is an isomorphism that can be constructed without any choices involved, like choosing a basis or an inner product or something of that nature to define the isomorphism.
Equality of 2 sets implies that the 2 sets contain the exact same elements.
My questions are particularly in regard to the natural isomorphism between a vector space ($V$) & its double dual ($(V^*)^*$) for finite dimensional vector spaces.
Question 1: We can prove that a vector space ($V$) is naturally isomorphic to its double dual double dual ($(V^*)^*$) but we cannot prove that they are equal, correct? Saying $V = (V^*)^*$ is an additional assertion that has nothing to do with the mathematics, right?
Question 2: If so what's the motivation for making this assertion? I understand that this allows us to think of vectors as (1,0) tensors but is there a deeper reason for doing this?
Question 3: Consider a real inner product space $(V, <.,.>)$ equipped with a special distinguished inner product, then a map $$ \phi : V \longrightarrow V^*$$ $$ \ \ : v \longrightarrow <., v>$$ then $\phi$ ought to be a natural isomorphism too, right? Does this mean that we can equate $V = V^*$ such that $v = \phi (v)$ & draw no distinction between vectors & covectors?