ca.analysis and odes - Can a conditionally convergent series of vectors be rearranged to give any limit?

Warmup (you've probably seen this before)

Suppose $sum_{nge 1} a_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to any real number $x$. To do this, let $P={nge 1|a_nge 0}$ and $N={nge 1|a_n<0}$. Since $sum_{nge 1} a_n$ converges conditionally, each of $sum_{nin P}a_n$ and $sum_{nin N}a_n$ diverge and $lim a_n=0$.

Starting with the empty sum (namely zero), build the rearrangement inductively. Suppose $sum_{i=1}^m a_{n_i}=x_m$ is the (inductively constructed) $m$-th partial sum of the rearrangement. If $x_mle x$, take $n_{m+1}$ to be the smallest element of $P$ which hasn't already been used. If $x_m> x$, take $n_{m+1}$ to be the smallest element of $N$ which hasn't already been used.

Since $sum_{nin P}a_n$ diverges, there will be infinitely many $m$ for which $x_mge x$, so $n_{m+1}$ will be in $N$ infinitely often. Similarly, $n_{m+1}$ will be in $P$ infinitely often, so we've really constructed a rearrangement of the original series. Note that $|x-x_m|le max{|a_n|bigm| nnotin{n_1,dots, n_m}}$, so $lim x_m=x$ because $lim a_n=0$.

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Suppose $sum_{nge 1}v_n$ is a conditionally convergent series with $v_nin mathbb R^k$. Can the sum be rearranged to converge to any given $win mathbb R^k$?

Obviously not! If $lambda$ is a linear functional on $mathbb R^k$ such that $sum lambda(v_n)$ converges absolutely, then $lambda$ applied to any rearrangement will be equal to $sum lambda(v_n)$. So let's also suppose that $sum lambda(v_n)$ is conditionally convergent for every non-zero linear functional $lambda$. Under this additional hypothesis, I'm pretty sure the answer should be "yes".