Weak Convergence in Hilbert Space/Corollary

Corollary
Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $\HH$.

Let $x \in X$.

Then:


 * $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$




 * $\innerprod y {x_n} \to \innerprod y x$ for each $y \in \HH$.

Proof
From Weak Convergence in Hilbert Space, we have:


 * $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$




 * $\innerprod {x_n} y \to \innerprod x y$ for each $y \in \HH$.

From Convergence of Complex Conjugate of Convergent Complex Sequence, we have:


 * $\sequence {x_n}_{n \mathop \in \N}$ converges weakly to $x$




 * $\overline {\innerprod {x_n} y} \to \overline {\innerprod x y}$ for each $y \in \HH$

From conjugate symmetry of the inner product, we have:


 * $\overline {\innerprod {x_n} y} \to \overline {\innerprod x y}$ for each $y \in \HH$




 * $\innerprod y {x_n} \to \innerprod y x$ for each $y \in \HH$.

Hence the result.