If $a(u,u) \leq \liminf a(u_n, u_n)$, is $a(u,u) \leq \liminf a(u_n, u_{n-1})$?

Question

Let $a\colon H \times H \to \mathbb{R}$ be some bilinear form and $H$ is a Hilbert space. If it is weak lsc, i.e., $$a(u,u) \leq \liminf a(u_n, u_n)$$ for a sequence $u_n \rightharpoonup u$ in $H$, then is $$a(u,u) \leq \liminf a(u_n, u_{n-1})$$ true? Or is there some name for this kind of property?

Answer 1

No, this is not true. Take $a$ to be the inner product and consider the sequence $$ e_1, -e_1, e_2, -e_2, \ldots, $$ where $(e_i)$ is an infinite orthonormal system.

I do not think that many bilinear forms $a$ have this property (unless there is some compactness hidden in $a$).