TITLE: Why conjugate when switching order of inner product?
QUESTION [15 upvotes]: There is an axiom of inner product spaces that states:

$\overline{\langle x,y\rangle } = \langle y,x\rangle$

Basically (without any conceptual understanding) it seems like all you have to do when you swap the order of the arguments in an inner product space is take their conjugate. 
How does this make any sense? I know if we are dealing with an inner product space over $\mathbb{R}$ then the conjugate of a real number is just the real number itself so there is no change. But how does this make sense over the field $\mathbb{C}$?

REPLY [4 votes]: The first thing to understand is how the definition of the inner product is slightly different in a complex context than the one you might have learned for the reals, in that you must take the conjugate of one of the arguments, so you have the vector definition:
$$
\langle x, y \rangle = x^*y
$$
or
$$
\langle x, y \rangle = x^Hy
$$
depending on your notation for the hermetian (conjugate) transpose.
Using the summation definition:
$$
\langle x, y \rangle = \sum_n \overline{x[n]}y[n]
$$
The reason we define it this way is that we want to be able to define the length $||u|| = \sqrt{\langle u, u \rangle}$, which should be a nonnegative real number. If we defined the inner product as $\langle x, y \rangle = \sum_n x[n]y[n]$, (or an integral for the continuous case) then any phase shift on elements of $u$ would be doubled. With the correct definition above, the phases will cancel out.
Another way to look at it (example from here) is to assume you have a non-zero vector $u$. To be a valid inner product, $\langle u, u \rangle$ should be a positive real number. So should $\langle iu, iu \rangle$, But with the normal real-valued inner product, $\langle iu, iu \rangle = i\langle u, iu \rangle = i^2\langle u, u \rangle = -\langle u, u \rangle < 0$, so this violates our condition.
so with the definitions above,$\langle y, x \rangle = y^*x = (x^*y)^* = \overline{\langle x, y \rangle}$
side note: this also this affects how you can pull coefficients out of an inner product:
$\langle cx, y \rangle = \overline c \langle x, y \rangle$
$\langle x, cy \rangle = c \langle x, y \rangle$
According to this, there's some disagreement about the argument ordering, but this seems consistent with the definition above.