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)


1 Answer 1


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.

  • 1
    $\begingroup$ (+1) ... And sometimes not even real, depending on the vector space it is acting on. $\endgroup$
    – Eff
    Mar 10, 2015 at 18:55
  • 1
    $\begingroup$ @Eff especially on a complex vector space (as OP has cited) (+1) $\endgroup$
    – AlexR
    Mar 10, 2015 at 19:09
  • $\begingroup$ Thanks for noting this, it's a good thing to include. $\endgroup$
    – BaronVT
    Mar 10, 2015 at 19:15

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