TITLE: Motive for the definition of inner product
QUESTION [12 upvotes]: Mathematicians pride themselves on writing proofs of propositions in an elegant way, but frequently (maybe even usually?) neglect to formally write motivations of definitions with the same elegance, efficiency, and sometimes beauty, and neglect to assign exercises in which the student is challenged to prove that a definition is the only (sometimes up to some logically equivalent formulations) one that satisfies some desiderata.  Often a textbook will just say dogmatically "A left and right conjugate sub-hypopotamus is a thing that blah blah blah${}\,\ldots$"  As I have remarked elsewhere in this forum, it is usually considered licit to define the concept of "group" by saying a group is a set with a binary operation satisfying etc. etc. etc. and then go on to prove a zillion theorems in group theory, rather than showing at the outset how the concept developed from a variety of concrete examples involving transformations in geometry, bijections, matrices, etc.
Definition: The inner product $\langle u,v\rangle$ of two vectors $u=(u_1,\ldots,u_n), v=(v_1,\ldots,v_n) \in\mathbb C^n$ is
$$
\sum_{k=1}^n u_k \overline{{}\,v\,{}}_k
$$
where $\overline{a}$ is the complex conjugate of $a\in\mathbb C$.
A friend once asked me why one takes conjugates.  And perhaps why only of the components of the second vector and not the first.
My answer was that it makes $\|u\|^2$ in every case positive (except when $u=0$) and makes the Pythagorean theorem work on neatly so that some basic Euclidean geometry can be applied in these spaces.  His reaction included a certain amount of indignation about not having been told that by textbooks and instructors.  Perhaps this bears the same relation to a good publication-worthy answer that a vague handwaving argument does to a proper proof.
So, suppose that our customs in regard to writing motivations of definitions were like those of our customs in regard to writing proofs.  We want them to be complete, correct, informative, satisfying to reasonable demands for justification, elegant, beautiful, comprehensible to the intended audience, and as simple as possible subject to the foregoing constraints.  Particularly clever motivations would be published in things like the Monthly in the same way that novel or unusually nice new proofs of old theorems are now published.  A particularly brilliant motivation for a new definition might be the whole of the topic of a paper in a research journal, maybe in rare cases winning a Fields medal.
So, how would one write a good publication-worthy motivation for the definition above in accordance with the standards outlined in the foregoing paragraph?

REPLY [3 votes]: Let us denote the canonical real inner product by $\langle v,w\rangle_{\mathbb{R}}$, where $v,w\in\mathbb{R}^n$.
If we want to define a complex inner product in $\mathbb{C}^n$, $\langle v,w\rangle_{\mathbb{C}}$, since $\mathbb{C}^n\supset\mathbb{R}^n$, we can think that this complex inner product is an extension of the real inner product.
So if $v,w\in\mathbb{R}^n$ then $\langle v,w\rangle_{\mathbb{C}}=\langle v,w\rangle_{\mathbb{R}}$.
Now, we should also ask that $\langle v,v\rangle_{\mathbb{C}}\geq 0$, because we want to define the norm of $v$ as $\|v\|=\sqrt{\langle v,v\rangle_{\mathbb{C}}}$.
Thus, we want two properties for $\langle v,w\rangle_{\mathbb{C}}$:

$\langle v,w\rangle_{\mathbb{C}}=\langle v,w\rangle_{\mathbb{R}}$, if $v,w\in\mathbb{R}^n$.
$\langle v,v\rangle_{\mathbb{C}}\geq 0$, because we want to define $\|v\|$ such that $\|v\|^2=\langle v,v\rangle_{\mathbb{C}}$.

Now, let us consider $n=1$. 
If $a,b\in\mathbb{R}^1$ then $\langle a,b\rangle_{\mathbb{C}}=\langle a,b\rangle_{\mathbb{R}}=ab$
If $a\in\mathbb{C}^1$ then $\langle a,a\rangle_{\mathbb{C}}$ must be the square of a norm. But $\mathbb{C}^1$ has a canonical norm, since we are trying to extend our definitions, let us use this canonical norm. So  $\langle a,a\rangle_{\mathbb{C}}=|a|^2=a\overline{a}$.
Thus, we obtained these two properties:

$\langle a,b\rangle_{\mathbb{C}}=ab$, if $a,b\in\mathbb{R}$.
$\langle a,a\rangle_{\mathbb{C}}=a\overline{a}$.

So how should be defined $\langle a,b\rangle_{\mathbb{C}}$ for $a,b\in\mathbb{C}^1$? Ans: $\langle a,b\rangle_{\mathbb{C}}=a\overline{b}$.
By analogy in $\mathbb{C}^n$, the inner product shoul be $\langle(a_1,\ldots,a_n),(b_1,\ldots,b_n)\rangle=\sum_{i=1}^na_i\overline{b_i}$ (Here we could also say that if $(V,\langle\cdot,\cdot\rangle_1)$ and $(W,\langle\cdot,\cdot\rangle_2)$ are inner product spaces then $V\times W$ is a inner product space with this inner product $\langle(v_1,w_1),(v_2,w_2)\rangle=\langle v_1,v_2\rangle_1+\langle w_1,w_2\rangle_2$).