I was going through the proof of triangle inequality as a consequence of Schwarz inequality here:
I find somethinng odd in the third step (the expansion of the inner product) . I learnt of the following properties of inner product:
$\langle a,b \rangle=\langle b,a \rangle^*$
$\langle a,zb+kc \rangle =z \langle a,b \rangle +k \langle a,c \rangle$
$ \langle za+kb,c \rangle =z^* \langle a,c \rangle + k^* \langle b,c \rangle$
$\langle za,kb \rangle = z^*k \langle a,b\rangle $
where $a,b,c$ are in general complex vectors, and $z,k$ are in general complex scalars.
But I think the third step is wrong, due to rule (1). We should have got $\langle y,x \rangle =\langle x,y \rangle^*$, and not $\langle y,x \rangle = \langle x,y \rangle$, unless both $x,y$ are taken to be real vectors. But should I think of it in this way: that the vectors involved in proving the triangle-inequality must be real, and can not at all possess an imaginary part?