Bessel's Inequality
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Theorem
Let $H$ be a Hilbert space, and let $E = \left\{{e_n: n \in \N}\right\}$ be a countably infinite orthonormal subset of $H$.
Then, for all $h \in H$, one has the inequality:
- $\displaystyle \sum_{n \mathop = 1}^\infty \left|{\left\langle{h, e_n}\right\rangle}\right|^2 \le \left\|{h}\right\|^2$
Corollary 1
If $E$ is an orthonormal subset of $H$, then for any $h \in H$, the set $\displaystyle \left\{{e_n \in E: \left\langle{h, e_n}\right\rangle \ne 0}\right\}$ 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): $I.4.8-10$
