Subspace $\ell^2$ is of first category

Question

Let $\ell^2$ be the Hilbert space of square summable sequences, and $\mathcal{H}$ be the subspace consisting of sequences $\{x_n\}$ with $\sum_{n=1}^\infty n^2|x_n|^2<\infty$
Show that $\mathcal{H}$ is of the first category.

I unsure how to start this problem.

Answer 1

We have to express $\mathcal H$ as a countable union of sets whose closure have an empty interior. Define $$ F_N:=\left\{ \left(x_n\right)_{n\geqslant 1}: \sum_{n=1}^{+\infty}n^2\left\lvert x_n\right\rvert^2\leqslant N\right\}. $$ If we manage to show that $F_N$ is closed and has an empty interior, then we are done. For the latter point, if we fix an element $x=\left(x_n\right)_{n\geqslant 1}\in\mathcal F_N$, then considering $y=\left(y_n\right)_{n\geqslant 1} \in\ell^2\setminus \mathcal H$, we can see that $x+\varepsilon y$ is not in $F_N$ for all positive $\varepsilon$.

Answer 2

Hint: Consider $E_M=\{(x_n)\in l^2: \sum n^2x_n^2\le M\}.$ Show $E_M$ is closed in $l^2.$ (Fatou's lemma). Then show $E_M$ has no interior.

Answer 3

With the obvious norm $\mathcal H$ is a Banach space. A version of the open mapping theorem says that if the range of a continuous linear operator is of second category then it is surjective and open. Apply this to the inclusion.