A key point is that -convergent- sequences are a closed subspace,
closed under multiplication, division (if non-zero), etc.

compare with $\bQ < \bR$: it's closed, while \emph{irrationals} are
not closed.

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series is tricky:
not closed under natural multiplications etc.:
- $l^\infty \times l^1 \to l^1$ works
(but don't even try $l^\infty \times $ convergent: take
 a non-absolutely convergent sequence, and it goes nuts!)
 ($l^1 < l^\infty$, so what you want makes sense...)
- abs. conv. convolve with any conv. seq. is okay

