Weak lower semicontinuity property of a bounded, coercive and linear operator

by StopUsingFacebook   Last Updated October 17, 2018 14:20 PM - source

Let $A\colon V \to V^*$ be a bounded linear coercive operator on a Hilbert space $V$.

Does it follow that for if $u_n \rightharpoonup u$ in $V$ (weak convergence) then $$\langle Au, u \rangle \leq \liminf \langle Au_n, u_n \rangle?$$

This is somehow related to weak lower semicontinuity of norms..?

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