Let \(V\) be an inner product space , and let \( S \subseteq V \) be an orthonormal system . Then an orthonormal basis \( S' \) exists such that \( S \subseteq S' \).
Proof
This can be proven using Zorn’s Lemma .
Consider
\[ P = \{ S \subseteq T : T \text{ is orthonormal system of } V \}. \]We need to show that every chain \( Q \subseteq P \) has an upper bound \(U\).
The upper bound is given by
\[ U = \bigcup_{T \in Q} T. \]By Zorn’s Lemma, we obtain \( S' \in P \) such that
\[ T \subseteq S' \quad \forall T \in P, \quad \text{and} \quad S \subseteq S'. \]