Nested subspaces in a Hilbert space Can one find an infinite decreasing chain $X_1\supset X_2\supset\dots$ of linear closed subspaces of $l_2$ such that $\bigcap X_i=\{0\}$, and for any $x\in l_2$ there exist $j$ such that $x\in X_j^{\perp}$?

If $X\subset l_2$ then $X^{\perp}=\{t|\forall x\in X (0=(t,x))\} $ is a closed linear subspace of $l_2$. If $S$ is a closed linear subspace of $l_2$ and $S\ne l_2$ then $int (S)=\phi.$ If $A\subset B\subset l_2$ then $B^{\perp}\subset A^{\perp}$. Hence,if $A_j \not \subset \{0\}$ for some $j$, then $int (A_k^{\perp})=\phi$ for all $k\geq j$ .But then  $$l_2=\cup_{k\in N}A_k^{\perp}=\cup_{k\geq j}A_k^{\perp}$$ which violates the Baire Category Theorem. So we must have $$A_j\subset \{0\} \text { for every } j.$$ The Baire Category Theorem : If $T$ is a complete metric space and $F$ is a countable family of closed nowhere-dense subsets of $T$ then $T\backslash \cup F$ is dense in $T$.In particular when $T \ne \phi$ we have $\cup F\ne T$....NOTES :(1)"c is nowhere-dense" means $int(\bar c)=\phi$.(2)By countable I mean not uncountable,that is,finite or countably infinite.
