= Positive definite symmetric operator
{title2=$\langle Lv,v\rangle>0$}
A <symmetric operator> $L$ on a real <Hilbert space> is strictly positive definite when $\langle Lv,v\rangle>0$ for every nonzero $v$ in its <operator domain>. Strict positivity proves uniqueness of a solution of $Lu=f$, but in infinite dimension it need not prove existence for every $f$. A <uniformly positive definite symmetric operator> additionally has a positive lower bound independent of $v$. Symmetry must be stated separately when positivity is defined only by a real quadratic expression: adding a <skew-symmetric matrix> changes the operator without changing that expression.
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