Find the number of interior points of this subspace of $l^2$.

Question

Consider the Hilbert Space $l^2$.

Let $S=\{(x_1,x_2,\cdot\cdot\cdot)\in l^2:\sum\dfrac{x_n}{n}=0\}$.

Then find the number of interior points of $S$.

Let $(x_1,x_2,\cdot,\cdot,x_n\cdot,\cdot)\in S^\circ$. Then we must have for some $r>0$ such that $(x_1,x_2,\cdot,\cdot,x_n\cdot,\cdot)+r\in S$ where $r=(r_1,r_2,\cdot,\cdot r_n,\cdot \cdot )$

But then $\sum \dfrac{(x_n+r_n)}{n}=\sum (\dfrac{x_n}{n}+\dfrac{r_n}{n})=\sum\dfrac{r_n}{n}$

Since $r>0\implies r_i>0 $ for at least one $i\implies \dfrac{r_i}{i}\neq 0\implies \sum \dfrac{r_i}{i}\neq 0$

Thus $S^\circ =\emptyset$

Is my answer correct? Please verify

Answer 1

As already mentioned in comments a more general result is true: Every proper subspace of a normed vector space has empty interior.

Perhaps you might try to prove more general result. It often happens that proof of a more general result can be simpler (clearer, bring more insight) than proof of a special case.

After showing this result it only remains to show that $S$ is a proper subspace of $\ell_2$. To see this, just notice that $(1,0,0,\dots)\notin S$ (and this sequence belongs to $\ell_2$).

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The only place in your proof (link to the current revision) which I think should deserve a bit more detailed explanation is this:

Then we must have for some $r>0$ such that $(x_1,x_2,\cdot,\cdot,x_n\cdot,\cdot)+r\in S$.

To explain why such $r$ exists we can simply take some $y$ such that $y\in \ell_2$ and $y>0$. (For example, $y_n=\frac1n$.)

If $x$ is an interior point, then there is an open ball $B(x,\varepsilon)\subseteq S$ for some $\varepsilon>0$. We simply take $r:=\frac\varepsilon2 \cdot r$.

Notice that the same argument would work with any $r\in \ell_2\setminus S$. (I.e., you do not necessarily need $r_i>0$ for each $i$. Although such element clearly does not belong to $S$.)

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Some stylistic advice (although this might be matter of taste, so maybe not everybody will agree): I consider the notation $x+r$ somewhat better. (Using similar symbols for similar objects.) So you could write at the beginning: "Let $x=(x_1,x_2,\cdot,\cdot,x_n\cdot,\cdot)\in S^\circ$" and then use simply the variable $x$ to denote this sequence.

Also you should not use the same symbol for two various things as you did in this comment where you have both sequence $r$ and radius $r$.

And perhaps, since quite often the letter $r$ is used for radius, you could have used a different letter instead of $r$ for the element of $\ell_2\setminus S$. (I guess this was the reason for two different $r$'s in the comment I linked to.)