TITLE: Can an inner product on a vector space be negative?
QUESTION [7 upvotes]: 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)

REPLY [18 votes]: 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.