Köthe–Toeplitz dual

Summary

The Köthe–Toeplitz duals of a sequence space $X$ are the sequence spaces of multipliers $a$ for which $\sum a_n x_n$ behaves well for every $x$ in $X$. The $\alpha$-dual asks for absolute convergence of $\sum a_n x_n$, and the $\beta$-dual asks for convergence. They play the role of a dual space that is again a space of sequences, without any topology. For example, the $\beta$-dual of $\ell^1$ is $\ell^\infty$, which matches the dual space of $\ell^1$ in functional analysis.