Let $\mathcal{H}$ be a separable Hilbert space and $\mathfrak{B}(\mathcal{H})$ be the banach algebra of bounded operators on $\mathcal{H}$. If $e$ is a unit vector in $\mathcal{H}$, is it possible to find unitary operators $U_n$ in $\mathfrak{B}(\mathcal{H})$ such that $U_n e$ form a maximal orthonormal set in $\mathcal{H}$? I think I need to use the axiom of choice to show this, but cannot find a way. Could anyone help me?
2026-03-25 09:35:42.1774431342
Finding a maximal orthonormal set in a Hilbert space
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Just extend $e$ to an orthonormal basis $\{e_n\}$ with $e_1=e$. There exist unitary operators $U_n$ such that $U_ne_1=e_n,U_ne_n=e_1$ and $U_ne_j =e_j$ for $j \notin \{1,n\}$. Of course $\{U_n e\} \equiv \{e_n\}$ is an orthonormal basis.