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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 !

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  • $\begingroup$ @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
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Hint

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 $x \in E$.

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  • $\begingroup$ 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

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