Let E be the set of real sequences $(x_n)_{n\in N}$ such that the series $\sum u_{n}^2$ converges.

For $x=(x_n)_{n \in N} $and $y=(y_n)_{n \in N}$

Inner product :

(x|y) = $\sum_{n=0}^{+\infty} x_ny_n$

Let F be the vector subspace of E formed from zero sequences starting from a certain rank.

If $ z=(\frac{1}{n+1})_{n\in N} $, show that $z \in E $and calculate $d(z,F)$

Showing that $z\in E$ is easy with Riemann. I tried to find an orthogonal base of F to calculate $d(z,F)$ but I couldn’t find one. Is there another way to do the calculus of a distance without using an orthogonal projection? Here the spaces considered have infinite dimension so there has to be another way !

  • $\begingroup$ @mathcounterexamples.net I am struggling with Latex as I am new to it and i still don’t know how to make a proper sum and can’t find help anywhere to achieve it. $\endgroup$
    – Julien
    Jan 24 at 17:43
  • $\begingroup$ @mathcounterexamples.net Should I repost my question as it had been downvoted ? $\endgroup$
    – Julien
    Jan 24 at 17:53
  • $\begingroup$ To what kind of Riemann theorem are you referring to? $\endgroup$ Jan 24 at 19:25
  • $\begingroup$ @mathcounterexamples.net there was a mistake in what i wrote. It should be clearer now $\endgroup$
    – Julien
    Jan 24 at 19:27


The given space $E$ is the Hilbert space $\ell^2(\mathbb R)$. The space $F$ of eventually vanishing sequences is usually denoted by $c_{00}$.

$c_{00}$ is dense in $\ell^2(\mathbb R)$. This is quite easy to prove. Hence $d(x,F)=0$ for any $x \in E$.

  • $\begingroup$ Could you give a hint to prove that $c_{00}$ is dense in $\ell^2(\mathbb R)$ ? $\endgroup$
    – Julien
    Jan 24 at 20:20
  • $\begingroup$ Look at the distance between an element $x \in E$ and the same element with zeros starting at rank $n$. $\endgroup$ Jan 24 at 20:26
  • $\begingroup$ For such a sequence of elements of $c00$, the distance intuitively tends to 0, thus proving that $x$ is adherent to $c00$ and that $c00$ is dense in $\ell^2(\mathbb R)$. I just don’t know how to formulate clearly with the inner product that is given $\endgroup$
    – Julien
    Jan 24 at 20:33
  • $\begingroup$ Is it correct to say that a following of followings tends to another following ? So that the distance tends to 0? $\endgroup$
    – Julien
    Jan 24 at 20:36
  • $\begingroup$ Just write what the distance is between the two elements I mentioned in the comment above. $\endgroup$ Jan 25 at 8:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.