Subspace $\ell^2$ is of first category 
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.  

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.
