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I have an orthonormal system of functions $$ U = \left\{ u_{\lambda}(x) \in L_{2}(\mathbb{R}_{+}) \mid \lambda \in \left\{-1,\ldots,-n\right\} \cup\mathbb{R}_{+} \right\} $$ such that for any $f,g \in L_{2}(\mathbb{R}_{+})$ we have $$ \langle f,g \rangle_{L_2} = \sum\limits_{k=-n}^{-1}\langle u_{k},f\rangle_{L_2} \overline{\langle u_k, g \rangle}_{L_2} + \int\limits_{0}^{\infty}\langle u_\lambda,f\rangle_{L_2} \overline{\langle u_{\lambda},g\rangle_{L_2}} d\lambda. $$ Is it possible to say something about completeness of $U$ in this case?

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up vote 2 down vote accepted

I don't believe that you can have an uncountable orthonormal system in a separable space.

Leaving that aside, one could argue as follows: if the closure of the span of $U$ is not all of $L_2$, then there is a nonzero element $f\in L_2$ orthogonal to it. Then $0\ne \langle f,f\rangle = \sum_{k=-n}^{-1} 0 + \int_0^\infty 0 =0$, a contradiction.

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