= Strict positivity without coercivity can fail variational solvability
On <l2 sequence space> $\ell^2$, let $(Lu)_n=u_n/n$ and $f_n=1/n$. This bounded <self-adjoint operator> has $\langle Lu,u\rangle>0$ for nonzero $u$, but the equation $Lu=f$ requires $u_n=1$, which is not in $\ell^2$. Taking the first $N$ coordinates equal to one makes $I(u)=-\sum_{n=1}^N1/n\to-\infty$. Thus strict positivity in isolation does not give existence in that <Hilbert space>; a <coercive bilinear form> and a continuous forcing functional in its norm are essential hypotheses for the usual minimization proof.
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