A canonical isomorphism from $V$ to $W$.

Question

I am reading "Linear Algebra" by Ichiro Satake.

The notion "canonical" is very difficult for me.
I have questions about canonical isomorphisms.

The author wrote as follows:

Taking the dual space, $V\mapsto V^*$, is a manipulation to produce a new vector space $V^*$ from the given one $V.$ What will then be obtained, if we repeat this process once again? By definition, $V^{**}$ is the vector space of all linear mappings of $V^*$ into $K.$ Now, for each $a\in V$, the correspondence $$V^*\ni f\mapsto f(a)\in K$$ is clearly a linear map from $V^*$ into $K.$ In fact, if we call this mapping $\phi_a$, then $\phi_a\in V^{**}$ and the mapping $V\ni a\mapsto\phi_a\in V^{**}$ gives an isomorphism of $V$ onto $V^{**}.$
This isomorphism is "canonical" in the sense that it is defined in a natural, intrinsic manner, independently of the choice of basis. So, in what follows, we shall identify $V$ with $V^{**}$ by this isomorphism: $$V=V^{**},$$ (where $x\in V$ is identified with $\phi_x\in V^{**}$).
For the iterated dual spaces, we have by $V=V^{**}$ $$V^{***}=(V^{*})^{**}=V^*,\\V^{****}=(V^{**})^{**}=V^{**}=V,\dots$$ which shows that no new spaces are obtained any more. In this sense, the operation $V\mapsto V^*$ is "reflexive."
Remark 1. Some readers might think that $V$ and $V^*$, both being $n$-dimensional vector spaces, can also be identified with each other, say by the isomorphism defined by the correspondence of the bases $(e_i)\leftrightarrow (f_i).$ However, this correspondence depends essentially on the choice of the bases and hence is not canonical. Although $V$ and $V^*$ are isomorphic, there are many choices for the isomorphism, and in general there is no way to select a canonical one from among them.

Let $V,W$ be $n$-dimensional vector spaces.

• If we define an isomorphism from $V$ to $W$ by using bases, is the isomorphism always non-canonical? (I cannot define "define an isomorphism from $V$ to $W$ by using bases".)
  Let $P(B_V, B_W)$ be a procedure which returns an isomorphism from $V$ to $W$, where $B_V$ is a basis of $V$ and $B_W$ is a basis of $W$ and $B_V$ and $B_W$ are inputs to $P$.
  Is there an example such that $P(B_V, B_W)=P(B_{V}^{'},B_{W}^{'})$ for any two bases $B_V$ and $B_{V}^{'}$ of $V$ and any two bases $B_W$ and $B_{W}^{'}$ of $W$?
• If we define an isomorphism from $V$ to $W$ without using bases, is the isomorphism always canonical?
• If we define two isomorphisms from $V$ to $W$ without using bases, are the two isomorphisms the same isomorphism?

Answer 1

Whether or not an isomorphism is canonical depends (loosely speaking) on whether it is defined without having to make arbitrary choices. So you could think of "noncanonical" as rewriting

this correspondence depends essentially on the choice of the bases and hence is not canonical

as

this correspondence depends essentially on the choice of [some structure or feature] and hence is not canonical

The arbitrary choice is often a choice of basis but need not be.

So for the first bullet: if the isomorphism actually depends on the bases then it's not canonical. The procedure $P$ you describe will be canonical if it behaves as you say. That can happen: the canonical isomorphism between a space and its second dual might be described that way (but that would be ugly and unnatural).

For the second and third bullets: perhaps. The answer depends essentially on whether you had to make any arbitrary choices.

If you go on to study more abstract algebra you will learn some category theory and encounter a more formal definition of "canonical".