Let \(H\) denote a Hilbert space , and let \( S \subseteq H \) be an orthonormal system . The following statements are equivalent [1, Satz 4.9]:

1. \( S \) is an orthonormal basis .
2. If \( x \in H \) and \( x \perp S \) then \(x=0\).
3. \( H = \overline{\operatorname{lin} S}\)
4. \( \forall x \in H: \quad x = \sum_{e \in S} \langle x, e \rangle e \)
5. \( \forall x, y \in H: \quad \langle x, y \rangle = \sum_{e \in S} \langle x, e \rangle \langle e, y \rangle \)
6. \( \|x\|^2 = \sum_{e \in S} |\langle x, e \rangle|^2 \) (Parseval’s equality)

References Link to heading

1. D. Werner, Funktionalanalysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. doi:10.1007/978-3-662-55407-4