(Weak) basis for the space of bounded sequences

Question

The canonical basis is not a Schauder basis of the space of bounded sequences, but in some way, it uniquely determines every element in the space. Is it a basis in a weaker sense? How is it called?

Thanks a lot.

Answer 1

A subspace of a normed vector space is closed if and only if it is weakly closed Closed $\iff$ weakly closed subspace. Hence, a set is a Schauder basis if and only if it is a "basis in the weaker sense".

Answer 2

Any sequence in $\ell^\infty$ can be understood as a operator of $\ell^1$ sequences. Then we can take a Schauder basis of $\ell^1$ and describe the behavior of the $\ell^\infty$ operator through the images of the basis elements.