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This may be a noob question but I recently read a definition that an inner product on a complex vector space is said to be a positive-definite sesquilinear map.

Doesn't positive definite mean that the inner product will only return positive values?

(Just started studying Functional Analysis specifically Hilbert Spaces)

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Positive-definite in this sense only means that $\langle x, x\rangle > 0$ if $x \neq 0$, it doesn't prohibit general $\langle x, y \rangle$ from being zero, negative, complex, etc.

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    $\begingroup$ (+1) ... And sometimes not even real, depending on the vector space it is acting on. $\endgroup$ – Eff Mar 10 '15 at 18:55
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    $\begingroup$ @Eff especially on a complex vector space (as OP has cited) (+1) $\endgroup$ – AlexR Mar 10 '15 at 19:09
  • $\begingroup$ Thanks for noting this, it's a good thing to include. $\endgroup$ – BaronVT Mar 10 '15 at 19:15

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