Why it is important for isomorphism between vector space and its double dual space to be natural?

Solution 1:

3) Of course you can define natural isomorphisms without using the language of category theory, but category theory was (in part) invented in order to efficiently express this notion, so there is little reason to (apart from trying to improve understanding).

2a) Philosophically, what makes an isomorphism between two objects natural is that constructing the isomorphism does not require more information than constructing the objects.

For example, in order to construct $V^*$ or $V^{**}$, it is enough to know that $V$ is a vector space. What I mean is that in order to build the two sets $V^*=\{f\colon V\to\mathbb F\mid f\text{ is linear}\}$, $V^{**}=\{f\colon V^*\to\mathbb F\mid f\text{ is linear}\}$, and to give them the structure of vector spaces, you need know nothing more than what a vector space is (hence what "linear" is), and that $V$ is a vector space.

Most likely in your book, to construct an injective linear map $\phi\colon V\to V^*$, you used additional information about the vector space $V$: probably a choice of basis. For $\phi$ to also be surjective, and hence an isomorphism, you need even more information: that the basis was finite*.

On the other hand, in order to construct the linear map $\psi_V\colon V\to V^{**}$, however, you don't need any extra information other than that $V$ is a vector space. You simply define $\psi_V(v)\in V^{**}$ by specifying how the $\psi(v)$ acts on elements $f\in V^*$: you declare $\psi_V(v)(f)=f(v)$. (To know that it is an isomorphism, you still need the extra information that $V$ is finite-dimensional, otherwise $\psi_V$ is just injective).

2b) Mathematically, the fact that an isomorphism is natural, i.e. does not depend on structural information beyond that contained in the objects, is captured by defining an isomorphism to be natural if its construction is preserved by structure-preserving (e.g. linear) maps. This is most easily expressed using the language of category theory (since the language of category theory was invented to express this).

Concretely, you can see that the isomorphism between $V$ and $V^*$ is not natural since the isomorphism based on $\{e_1,\dots,e_n\}$ likely defines the dual basis $e_i^*(\sum a_je_j)=a_i$. But all this is doing is defining an inner-product (non-degenerate bilinear form if the base field $\mathbb F\neq\mathbb R$) $\left<\cdot,\cdot\right>$ on $V$ such that $\left<e_i,e_j\right>=\delta_{ij}=\begin{cases}0&i\neq j\\1&i=j\end{cases}$. The basis (when $\mathbb F=\mathbb R$) allows us to identify $V$ as our usual $n$-dimensional space with coordinates, the inner product becomes the dot product, and the isomorphism to $V^*$ sends a vector $\vec v$ to the functional that projects orthogonally onto $\vec v$. Then the only linear transformations of $V$ to itself that preserve the isomorphism to $V^*$ are the orthonormal transformations, ones that preserve the length and (unsigned) angle between vectors. All other transformations of $V$ to itself break the isomorphism, and hence we can conclude the isomorphism is not natural.

1) The use of natural transformations is that they ensure that the maps you are constructing reflect genuine properties of the mathematical object, rather than being an artifact of the specific way in which you present the object, and hence a consequence of properties that the object "in-and-of-itself" does not have.

*More generally, the extra information for $\phi$ that you need is that of a bilinear form, which the basis allows you to define, and for $\phi^{-1}$ you need the bilinear form to be non-degenerate, which is what finiteness of the basis allows you to build. However, there are other ways to build non-degenerate bilinear forms, e.g. $L^2$ inner products.

EDIT: in response to comments, note that it makes no sense to talk about natural isomorphisms between "unnatural constructions". I said that philosophically, a natural construction is one that depends on no extra information. Mathematically, it is a functor: in addition to telling us how to construct a new object $F(V)$ out of an old object $V$, a natural construction also tells us how the construction behaves if we are given a structure-preserving map $f\colon V\to W$, by giving a structure-preserving map $F(f)$ between $F(V)$ and $F(W)$.

There are two types of natural constructions: covariant and contravariant, depending on whether $F(f\circ g)=F(f)\circ F(g)$ or $F(f\circ g)=F(g)\circ F(f)$. In particular, one shows the construction of $V^*$ is (contravariantly) natural by defining/constructing, for any $f\colon V\to W$, an $f^*\colon W^*\to V^*$ given by $f^*\colon g\mapsto g\circ f$ for any $g\colon W\to\mathbb F$, and showing that $(f_1\circ f_2)^*(g)=g\circ f_1\circ f_2=(f_2^*\circ f_1^*)(g)$.

Now, a natural transformation between $I$ (the identity construction $I(V)=V$ and $I(f)=f$ are isomorphisms $\phi_V\colon I(V)\to F(V)$ such that: $ \require{AMScd} \begin{CD} I(V) @>{\phi_V}>> F(V)\\ @VfVV @VVF(f)V \\ I(W) @>{\phi_W}>> I(W) \end{CD}$ for covariant $F$ or $\begin{CD} I(V) @>{\phi_V}>> F(V)\\ @VfVV @AAF(f)A \\ I(W) @>{\phi_W}>> I(W) \end{CD}$ for contravariant $F$. Now it is easy to see that there is no natural isomorphism between $V$ and $V^*$. For suppose that there were, so that we have $\begin{CD} V @>{\phi_V}>> V^*\\ @VfVV @AAf^*A \\ W @>{\phi_W}>> W^* \end{CD}$ for every $f\colon V\to W$. Then clearly, $\phi_V^{-1}\circ f^*\circ\phi_W\circ f$ would be the identity, so $f^{-1}\colon W\to V$ would be given by $\phi_V^{-1}\circ f^*\circ\phi_W$. But not every linear map is an isomorphism (in particular, the $f(v)=0$ is a linear map that is not an isomorphism for any choice of $V$ and $W$), so contradiction.