# Why is this true? (standard a-priori estimate on linear parabolic PDE)

We see that $$\frac{1}{2}\lVert u(T) \rVert^2_{L^2} \leq \frac{1}{2}\lVert u(0) \rVert^2_{L^2} + \lVert f \rVert_{L^2(0,T;H^{-1})}\lVert u \rVert_{L^2(0,T;H^1_0)}$$ and since $T$ is arbitrary, we have $$\lVert u \rVert^2_{L^\infty(0,T;L^2)} \leq \lVert u(0) \rVert^2_{L^2} + \lVert f \rVert^2_{L^2(0,T;H^{-1})}$$