Is the inner product $\langle \cdot, \cdot\rangle$ lower semi-continuous w.r.t. the product of weak topologies?

Question

Let $(H, \langle \cdot, \cdot\rangle)$ be a real Hilbert space. Let $\sigma (H, H^*)$ be the weak topology of $H$. Let $\tau :=\sigma (H, H^*) \otimes \sigma (H, H^*)$ be the product topology of $\sigma (H, H^*)$ and itself. Then $\tau = \sigma (H^2, (H^2)^*)$.

Is the inner product $\langle \cdot, \cdot\rangle:H^2 \to \mathbb R$ lower semi-continuous w.r.t. $\tau$?

Thank you so much for your elaboration!

Answer 1

It is not lower semicontinuous unless $H$ is finite-dimensional. If it were the case, then $u \mapsto \langle u,-u\rangle = -||u||^2$ would be weakly lower semicontinuous on $H$, which is not true.