Orthonormal Sequence in Hilbert Space Converges Weakly to Zero

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

Let $\sequence {e_n}_{n \mathop \in \N}$ be a sequence such that:


 * $\innerprod {e_n} {e_m} = 0$ if $n \ne m$

and:


 * $\norm {e_n} = 1$ for each $n \in \N$.

Then:


 * $e_n \rightharpoonup 0$

where $\rightharpoonup$ denotes weak convergence.

Proof
From Bessel's Inequality, we have:


 * $\ds \sum_{n \mathop = 1}^\infty \size {\innerprod y {e_n} }^2$ converges for each $y \in \HH$.

So from Terms in Convergent Series Converge to Zero, we have:


 * $\cmod {\innerprod y {e_n} }^2 \to 0$ for each $y \in \HH$.

We show that:


 * $\innerprod {e_n} y \to 0$

Let $\epsilon > 0$.

Then we can find $n \in \N$ such that:


 * $\cmod {\innerprod y {e_n} }^2 < \epsilon^2$

for $n \ge N$.

Then we have:


 * $\cmod {\innerprod {e_n} y} = \cmod {\innerprod y {e_n} } < \epsilon$

for $n \ge N$.

Since the inner product is conjugate symmetric, we have:


 * $\innerprod y {e_n} = \overline {\innerprod {e_n} y}$

so:


 * $\cmod {\innerprod {e_n} y} < \epsilon$

for $n \ge N$, from Complex Modulus equals Complex Modulus of Conjugate.

So we have:


 * $\innerprod {e_n} y \to 0$

Since $y$ was arbitrary, we then have:


 * $e_n \rightharpoonup 0$

from Weak Convergence in Hilbert Space.