What does the inner product of two complex general vectors have to do with complex conjugation? $
\newcommand{\qr}[1]{|#1 \rangle} 
\newcommand{\ql}[1]{\langle #1|} 
\newcommand{\q}[2]{\langle #1 | #2 \rangle}
\newcommand{\v}[2]{\langle #1,#2\rangle}
$ I'm reading "Quantum computer science, an introduction," N. David Mermin, Cambridge University Press, 2007.  In Appendix A, he writes that "in a vector space over the complex numbers the inner product of two general vectors is a complex number satisfying $\q{\psi}{\phi} = \q{\psi}{\phi}^*$, where $*$ denotes complex conjugation." (Page 160.)
I must not have understood this.  I tried the following example.  What's the inner product of $\v{2+3i}{7+3i} \cdot \v{1+2i}{2+2i}$?  It's $4 + 27i$.  If I commute $\v{1+2i}{2+2i} \cdot \v{2+3i}{7+3i}$, I get the same $4 + 27i$ because inner products are commutative.  Taking the complex conjugate $4 + 27i$, I get $4 - 27i$ which is not the same as its complex conjugate, $4 + 27i$.
So I'm pretty lost here.  What does he mean by what he said?
 A: Inner product on comlex vetorspace have conjugate symmetry (so not commutative).
An inner product is a function that assigns to every ordered pair of vectors $x$ and $y$ in V a scalar in $\mathbb{C}$, denoted $  \langle x,y\rangle$, such that  $\forall\ x, y, z \in V $ and $ c \in \mathbb{C} $:
\begin{equation*}
V \rightarrow F: \quad x,y \rightarrow \langle x,y\rangle
\end{equation*}
\begin{array}
((a)\quad  \langle x+z,y\rangle= \langle x,y\rangle+ \langle z,y\rangle \\[5pt]   
(b)\quad  \langle cx,y\rangle= c \langle x,y\rangle  \\[5pt] 
(c)\quad  \overline{\langle x,y\rangle} =\langle y,x\rangle    \\[5pt] 
(d)\quad  \langle x,x\rangle >0 \  \text{if} \  x\ne 0  \\[5pt] 
\end{array}
Here $\overline{x}$ denotes the complex conjugate of $x$.
In complex n dimensional euclidean space the inner product of two vectors $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ defined by $\sum_{j=1}^n \overline{a_j}b_j$
So in your example the correct way of calculation is
\begin{equation}
\langle{2+3i},{7+3i}\rangle \cdot \langle{1+2i},{2+2i}\rangle=28+9i
\end{equation}
and
\begin{equation}
\langle{1+2i},{2+2i} \rangle \cdot \langle{2+3i},{7+3i}\rangle =28-9i
\end{equation}
which is consistent with  $(c)\  \overline{\langle x,y\rangle} =\langle y,x\rangle$.
A: To a physicist, the inner product of vectors $(a_1,\ldots,a_n)$
and $(b_1,\ldots,b_n)$ is $\sum_{j=1}^n \overline{a_j}b_j$. Also
a physicist would write $z^*$ for the complex conjugate of $z$
rather than $\overline z$.
