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?

Thanks in Advance


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.

  • 2
    $\begingroup$ Notice that to speak of a Hilbert basis you need a topology on your space... $\endgroup$ – Emanuele Paolini Nov 2 '13 at 9:18
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    $\begingroup$ Of course, it's implicit mentioned in the \overline. $\endgroup$ – Michael Hoppe Nov 2 '13 at 11:08
  • $\begingroup$ 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. $\endgroup$ – polynomial_donut May 25 '18 at 14:41
  • $\begingroup$ I wrote in my first sentence how a “usual” basis is defined. $\endgroup$ – Michael Hoppe May 25 '18 at 17:22

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