A real <linear operator> $\mathcal L$ on an <inner-product space> is a <positive-definite operator> when it is self-adjoint and
$$
\langle\mathcal Lu,u\rangle>0
$$
for every nonzero $u$ in its domain. In the variational setting one normally requires the stronger uniform estimate $\langle\mathcal Lu,u\rangle\geq c\|u\|_V^2$ for some $c>0$, which is <coercivity>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 341 / Section A / 4 / a / Solution

A self-adjoint operator $L$ is positive definite when $\langle Lu,u\rangle>0$ for every nonzero vector $u$ in its domain. For variational problems one usually asks for the uniform lower bound $\langle Lu,u\rangle\geq c\|u\|^2$ called <coercive bilinear form>[coercivity].

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