Prove a lower bound of the smooth convex function with Lipschitz continuous gradient how to prove this conclusion:

Suppose that $f$ is a smooth convex function with Lipschitz continuous gradient on $X$, then there exists a self-adjoint and positive semi-definite linear operators $\sum_{f}$ such that for any $x, x^{'}\in X$,
$$f(x)\geq f(x') + \langle x -x', \nabla f(x')\rangle + 0.5 *\|x-x'\|_{\sum_{f}}^2.$$

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What about the function $f(x) = 0$ !!!
What about the function $f(x) = 0$ !!!