Metrics on Set of Sequences of Real Numbers

[BullCityFats]

Let S be the set of sequences of real numbers with finite non-zero values.

Let f:S->R such that f(s) is the sum of the elements of S.

Does there exist a metric d on S such that

a) f is continuous; and
b) the dual space (S,d)* contains the set of convergent series.

I've been out of academia for a couple of decades so my Banach spaces are a little rusty, but
I'm inclined to say there is not. For d = max(|a_i-b_i|), f is not continuous; for d = sum(|a_i-b_i|),
the dual space does not contain all convergent series; for d = (sum((|a_i-b_i|)^n))^(1/n), neither
condition is true. But I can't rule out the possibility of some non-standard metric fulfilling my conditions.

Yes, I know I should define S as a vector space and d as a norm rather than a metric, but whatever.

Thanks in advance!