= Solution
Define the <continuous linear functionals> $f_n(z)=\langle x_n,z\rangle$. For each fixed $z\in X$, <weak convergence> makes $f_n(z)$ a convergent, hence bounded, scalar sequence. The <Uniform boundedness principle> states that a pointwise bounded family of <bounded linear operators> from a <Banach space> to a <normed space> has uniformly bounded <operator norms>. Since a <Hilbert space> is a <Banach space>, it applies to $(f_n)$.
The <Cauchy-Schwarz inequality> gives $\|f_n\|\leq\|x_n\|$, and evaluation at $z=x_n/\|x_n\|$ gives equality when $x_n\ne0$; equality also holds at zero. Consequently
$$
\boxed{\sup_n\|x_n\|=\sup_n\|f_n\|<\infty.}
$$
Back to article page
