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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

LikeMath

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

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