I am very confused whether or not the set
$G=\{(x_n)_{n\in\mathbb{N}} \in \ell_2: \sum_{n=1}^{\infty}\frac{x_n}{\sqrt{n}}=0\}$
is closed in $\ell_2$. Any hint is appreciated!
Asked
Active
Viewed 280 times
1
Daniel
- 5,872
-
The specification of $G$ is incomplete, are you missing an order of some sort? – copper.hat Apr 19 '20 at 19:33
-
To clarify copper.hat's issue, you the criterion for $x \in G$, is incomplete, $\sum_{n=1}^\infty \frac{x_n}{\sqrt{n}}$ is not a logical statement. – Physical Mathematics Apr 19 '20 at 19:35
-
That's all I was given, thus I am very confused – Apr 19 '20 at 19:35
-
I fixed it thanks! – Apr 19 '20 at 19:36
-
How is that sequence a counterexample? – Physical Mathematics Apr 19 '20 at 19:50
-
1Also why is your title about compactness but your question about closedness? These are not the same. – Physical Mathematics Apr 19 '20 at 19:52
-
Compactness and Closed is the same, I am fixing it just in case – Apr 19 '20 at 19:55
-
Compactness and closedness are *not* the same in general. In metric spaces this is true iff the space is compact, which your space is not. – Physical Mathematics Apr 19 '20 at 20:02
-
$l_2$ is already a metric space from what have learned – Apr 19 '20 at 20:38
-
I don't think that the set $G$ is well defined. I believe (based on a reading of https://math.stackexchange.com/a/99213/27978) that $\sum_k {x_k \over \sqrt{k}}$ is not defined for all of $l_2$. – copper.hat Apr 20 '20 at 02:07
1 Answers
1
Note that $(\frac{1}{\sqrt{2}},-1,0,\dots), (\frac{1}{\sqrt{3}},0,-1,0,\dots), \ldots $ belong to $G$. Moreover, note that $G\neq\ell_2$.
Let us assume that $G$ is closed. Since $G\neq\ell_2$, there is a non-null vector $\left(x_n\right)_{n\in\mathbb{N}}$ orthogonal to $G$. Therefore $(x_n)$ is orthogonal to each sequence above.
Thus, $\frac{x_1}{\sqrt{2}}-x_2=0$, $\ \frac{x_1}{\sqrt{3}}-x_3=0$, $\ \ldots.\ $ So $x_n=\frac{x_1}{\sqrt{n}}$.
Note that $(x_n)_{n\in\mathbb{N}}=\left(\frac{x_1}{\sqrt{n}}\right)_{n\in\mathbb{N}}$ belongs to $\ell_2$ if only if $x_1=0$. Absurd!
Daniel
- 5,872