The following definition can be found in the book: Friedberg et. al., Linear Algebra, 5th Edition, page 327.
Definition. Let $V$ be a vector space over $F$. An inner product on $V$
is a function that assigns, to every ordered pair of vectors $x$ and $y$ in $V$, a scalar in $F$ , denoted $\langle x,y \rangle$, such that for all $x$, $y$, and $z$ in $V$ and all $c$ in $F$, the following hold:(a) $\langle x + z, y \rangle = \langle x,y \rangle + \langle z,y \rangle$
(b) $\langle cx, y \rangle = c \langle x, y \rangle$
(c) $\overline{\langle x, y \rangle } = \langle y,x \rangle$, where the bar denotes complex conjugation
(d) If $x \neq \overrightarrow{0}$, then $\langle x, x \rangle$ is a positive real number
In this definition, is there a bar missing in the RHS of c)?
If not, what is the idea behind this property c) which we require here (as an inner product space axiom, if I may call it that way)? Is there any example or geometric meaning or anything else, so that one can build an intuition for c)?
Initially I thought yes (there's a bar missing in the RHS), but then I found the same equality in other sources/texts. So it seems the book definition is OK, it's not a typo.
Note
I've never studied inner product spaces before, the furthest I went
was dot/scalar product spaces, I don't think this axiom/property c)
is present there.
Best Answer
The complex conjugation in (c) leads to the definition $$ \langle x, y \rangle = \sum_i x_i \bar {y_i} $$ on $\mathbb{C}^n$.
That definition makes $\langle x, x \rangle \ge 0$, which is very useful for an inner product. When $n=1$ it is the square of the length of $x$ in the complex plane.