An inner product on a real or complex vector space is a scalar-valued function satisfying additivity, $\langle ax,y\rangle=\overline a\langle x,y\rangle$, linearity in the second slot, conjugate symmetry, and positive definiteness: $\langle x,x\rangle\ge0$ with equality only at $x=0$. Over real scalars, conjugation disappears and the function is symmetric bilinear.
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