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If $\{\frac{f_{n}}{\|f_{n}\|}\}_{n\in I}$ is an orthonormal basis for a separable Hilbert space $H$, and $\{f_{n}\}_{n\in I}$ is a complete and orthogonal set in $H$, is it true that $\{f_{n}\}_{n\in I}$ is a basis for $H$?

If this is not always true, when it would be?

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    @Liza: So if you are meaning to ask, "Is the smallest closed linear subspace of $H$ containing $\{f_n\}$ equal to $H$?" then the answer is "Yes." Such a subspace also contains an orthonormal basis $\{f_n/\|f_n\|\}$ for $H$. It is still not 100% clear to me what your question is asking. The quote you gave never precisely defined "orthogonal basis," but my guess is you meant what I wrote above, which in this case is equivalent to whether it is a Schauder basis as mentioned in David Mitra's comment.2012-05-28

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It is true that $\{f_n\}$ is a Schauder basis for $H$, which by definition means that every $x\in H$ has a unique representation as $\sum_{n=1}^\infty \alpha_n f_n$. Indeed, it is a direct consequence of the definition that multiplying each vector of a Schauder basis by a nonzero scalar gives another Schauder basis. (Based on a comment by David Mitra.)