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 $x$

if and only if:

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

Proof

From Weak Convergence in Hilbert Space, we have:

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

if and only if:

$\innerprod {x_n} y \to \innerprod x y$ 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 $x$

if and only if:

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

if and only if:

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

Hence the result.

$\blacksquare$