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A Hilbert space E is spanned by a set S if E is generated by the element of S.

It is well known that in the finite dimensional case that

S spans E and S is linearly independent set iff the set S form a basis for E.

The question is that true for the infinite dimensional case? Noting that in this case by span we mean the closure of the span.

Thanks in advance

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# Does every independent set spans the space necessarily form a basis?

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