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Can anybody give an example of a Euclidean Space, $E$, and an orthonormal system $\{x_n\}$ such that $E$ contains no nonzero element orthogonal to every $x_n$, even though $\{x_n\}$ fails to be complete.

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I'm not sure what definition of a complete orthonormal system you have, but there are two possibilities that I can think of:

Let $\{ x_i\, :\, i \in I \}$ be an orthonormal system in a Hilbert space $X$.

Definition 1. $\{ x_i \}$ is complete if, for each $x \in X$, $\{ x_i\,:\, i \in I \} \cup \{ x \}$ is not an orthonormal system.

Definition 2. $\{ x_i \}$ is complete if the closed linear span of its elements is $X$.

If you're asking this question, my guess is that you've been shown definition 2. But they are equivalent.

To prove that $(1) \Rightarrow (2)$, if $x \in X$ then we must have $\langle x, x_i \rangle = 0$ for each $i \in I$, since otherwise we could append $x$ and have an orthonormal system, and so $\overline{\text{span}\, \{ x_i \}} = X$.

To prove that $(2) \Rightarrow (1)$, if $\overline{\text{span}\, \{ x_i \}} = X$ then $\{ x_i \}^{\perp} = X^{\perp} = \{ 0 \}$, so no element of $X$ is orthonormal to each $x_i$.