Prove that $\text{Re} B(\mathcal{H})=\{T\in B(\mathcal{H}): T=T^* \}$ is WOT closed in $B(\mathcal{H})$

Question

Let $\mathcal{H}$ be a Hilbert space , Prove that $\text{Re} B(\mathcal{H})=\{T\in B(\mathcal{H}): T=T^* \}$ is WOT closed in $B(\mathcal{H})$

I will explain my attempt:

Let $(T_i)$ be a net in $B(\mathcal{H})$ such that $T_i \xrightarrow{WOT} T$ that is for all $x,y \in B(\mathcal{H})$, $\langle T_ix,y\rangle \to \langle Tx,y\rangle$

I am not sure how to verify this $\langle T^*_ix,y\rangle \to \langle T^*x,y\rangle$

thanks to everyone for answers

Answer 1

This follows immediately from the definition of adjoint: $ \langle T^{*}x, y \rangle=\langle x, Ty \rangle$ which is the complex conjugate of $\langle Ty, x \rangle$ and similarly for $T_i$.