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 in $\HH$ such that:

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

and:

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

Then:

$e_n \weakconv 0$

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 then have, from Complex Sequence is Null iff Positive Integer Powers of Sequence are Null:

$\cmod {\innerprod y {e_n} } \to 0$ for each $y \in \HH$

and so, from Complex Sequence is Null iff Modulus of Sequence is Null:

$\innerprod y {e_n} \to 0$ for each $y \in \HH$.

From Weak Convergence in Hilbert Space: Corollary, we therefore have:

$e_n \weakconv 0$

$\blacksquare$

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $27.1$: Weak Convergence