Weak lower semicontinuity

Remarks

Replacing sequences by the weak topology itself gives weak lower semicontinuity in the topological sense, which implies the sequential form. In the calculus of variations the sequential form is the one that is used.

Example: the norm $J(u) = \norm{u}$ is weakly lower semicontinuous. By the Hahn–Banach theorem $\norm{u} = \sup \abs{\ell(u)}$ over continuous linear $\ell$ with $\norm{\ell} \le 1$. For each such $\ell$, $\abs{\ell(u)} = \lim_k \abs{\ell(u_k)} \le \liminf_k \norm{u_k}$.

Non-example: on $X = L^2(0, 2\pi)$ let $J(u) = -\norm{u}^2$ and $u_k(x) = \sin(kx)$. Then $u_k \to 0$ weakly, by the Riemann–Lebesgue lemma, and $J(u_k) = -\pi$ for all $k$. But $J(0) = 0 > -\pi$, so $J$ is not weakly lower semicontinuous, although it is continuous for the norm.

By Mazur's lemma, a convex functional that is lower semicontinuous for the norm is weakly lower semicontinuous.