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Let H denote Hilbert space, the space of square-summable infinite sequences of real numbers-which is infinite-dimensional and separable. Let S1,S2 denote subsets of H such that a point p of H belongs to S1 or S2 according to whether the sum of the abslolute values of the co-ordinates of p is convergent or divergent respectively. Has the topology of these subsets of H been much investigated? It is clear that the positive multiples of any point p of H remain in the same set (S1 or S2) that p itself belongs to. Since the origin of H belongs to S1 it is clear that S1 is arcwise connected. But is S2 also arcwise connected? I can see no easy way to prove this. Each point of H is a limit point of S1 and also a limit point of S2. Hence neither of these sets is either an open or a closed subset of H. Is it possible that S1 and S2 are not even Borel subsets of H? I started thinking about these sets in the course of trying to come up with a series of positive rational numbers whose convergence or divergence might be undecidable in Peano's Arithmetic or even in ZFC.

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up vote 6 down vote accepted

The unit ball of $\ell_1$ is weakly compact in $\ell_2$ and closed in $\ell_2$, so $S1$ is $F_\sigma$ and $S2$ is $G_\delta$.

ADDED 10/22/11: $S2$ is also arcwise connected. Give $x_i$ in $S2$ for $i=1,2$ you can choose a partition $A \cup B$ of the natural numbers so that $1_A x_i$ and $1_B x_i$ are in $S2$ for $i=1,2$. The straight line paths from $x_1$ to $1_A x_1 +1_Bx_2$ and from $1_A x_1 +1_Bx_2$ to $x_2$ stay in $S2$.

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Thanks alot for providing such clear and complete answers to all of my questions. Your proof of the arcwise connectedness of S2 is really neat. –  Garabed Gulbenkian Oct 24 '11 at 22:07
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