An inner product $\langle \cdot,\cdot \rangle$ satisfies the following properties:
Let $u$, $v$, and $w$ be vectors and $\alpha$ be a scalar, then:
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$\langle u+v,w\rangle=\langle u,w\rangle+\langle v,w\rangle$ and $\langle \alpha v,w\rangle=\alpha\langle v,w\rangle$ (Linearity in first coordinate).
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$\langle v,w\rangle=\langle w,v\rangle$ (Symmetry) .
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$\langle v,v\rangle\ge 0$ and $\langle v,v\rangle=0$ $\Leftrightarrow$ $v=0$ (Positive definiteness).
This the general definition of inner products, but in complex vector spaces the symmetry part is $\langle u,v\rangle=\overline{\langle v,u\rangle}$, how can be possible? this is supposed to be the same symmetry of the general properties above, btw what's the difinition of $\overline{\langle u,v\rangle}$?
EDIT
I know how to prove $\langle u,v\rangle=\overline{\langle v,u\rangle}$, what I wanna know is why this is called an inner product, because we don't have $\langle u,v\rangle={\langle v,u\rangle}$
Thanks in advance
Best Answer
There is no way to preserve all of these 3 properties in complex field. We have to choose the least important one to sacrifice.
Linearity is the foundation of linear algebra, and positive definiteness is required to make inner product a metric. Therefore, we abandon symmetry.