# Algebraic Basis vs Hilbert basis

I am confused between algebraic basis and hilbert basis. How do they differ exactly? Can you give me examples (possibly in infinite dimensions) on when they are the same and when they are not the same?

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, an algebraic basis is a more specific term for the $usual$ basis taught in basic courses on linear algebra? Asking, because I've never heard about algebraic bases. – polynomial_donut May 25 '18 at 14:41