Algebraic Basis vs Hilbert basis

real-analysis linear-algebra

A basis $B$ of a vector space $V$ allows you to express all vectors as a finite sum with vectors of that base. That is: $V=\mathrm{span}(B)$

In case of a Hilbert-basis all vector are expressed with a (maybe) infinite sum of vectors of that Hilbert-basis. And that is: $V=\overline{\mathrm{span}(B)}$.

So in that sense a Hilbert-basis is not a basis.

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