Bessel's Inequality
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Theorem
Let $H$ be a Hilbert space.
Let $E = \set {e_n: n \in \N}$ be a countably infinite orthonormal subset of $H$.
Then, for all $h \in H$:
- $\ds \sum_{n \mathop = 1}^\infty \size {\innerprod h {e_n} }^2 \le \norm h^2$
Corollary 1
If $E$ is an orthonormal subset of $H$, then for any $h \in H$, the set $\ds \set {e_n \in E: \innerprod h {e_n} \ne 0}$ is countable.
Corollary 2
The condition in the theorem that the orthonormal set $E$ be countable is superfluous.
Proof
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Source of Name
This entry was named for Friedrich Wilhelm Bessel.
Sources
- 1990: John B. Conway: A Course in Functional Analysis ... (previous) ... (next): $\text I.4.8-10$