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We should not forget the famous Riesz-Fischer Theorem:

Let $H$ be a Hilbert space and let ${u_{\alpha}:\alpha\in A}$ be an orthonormal set in $H$. Suppose $\phi$ is in the $l^2$-space of $(A,\mu)$ where \mu $\mu$ is the counting measure on $A$. Then $\phi=\widehat{x}$ for some $x\in H$ where $\widehat{x}:A\to \mathbb{C}$ is defined by $\widehat{x}(\alpha)=(x,u_{\alpha})$, the inner product of $x$ with $u_{\alpha}$, for each $\alpha\in A$.

(I quote Theorem 4.17, page 89, in the second edition of Walter Rudin's Real and Complex Analysis.) In fact, I do not think it would be an exaggeration to say that the Riesz-Fischer Theorem is nothing but a reformulation of the completeness of $H$ - that is how crucial the assumption of completeness is to the proof.

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We should not forget the famous Riesz-Fischer Theorem:

Let $H$ be a Hilbert space and let ${u_{\alpha}:\alpha\in A}$ be an orthonormal set in $H$. Suppose $\phi$ is in the $l^2$-space of $(A,\mu)$ where \mu is the counting measure on $A$. Then $\phi=\widehat{x}$ for some $x\in H$ where $\widehat{x}:A\to \mathbb{C}$ is defined by $\widehat{x}(\alpha)=(x,u_{\alpha})$, the inner product of $x$ with $u_{\alpha}$, for each $\alpha\in A$.

(I quote Theorem 4.17, page 89, in the second edition of Walter Rudin's Real and Complex Analysis.) In fact, I do not think it would be an exaggeration to say that the Riesz-Fischer Theorem is nothing but a reformulation of the completeness of $H$ - that is how crucial the assumption of completeness is to the proof.