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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-341/section-a/4/a/solution