Köthe-Toeplitz dual

For any subset $X$ of the set $\omega$ of all sequences $x = (x_k)$, the set $$ X^\alpha = \{ a \in \omega : \sum_k | a_k x_k | < \infty \ \text{for all}\ x \in X \} $$ is called a Köthe–Toeplitz or $\alpha$-dual of $X$. These duals play an important role in the representation of linear functionals (cf. Linear functional) and the characterization of matrix transformations between sequence spaces. They are special cases of the more general multiplier sequence spaces $$ Z = M(X,Y) = \{ a \in \omega : ax = (a_k x_k) \in Y \ \text{for all}\ x \in X \} $$ which for $Y = \mathrm{cs}$ and $Y = \mathrm{bs}$, the sets of convergent or bounded series, reduce to $X^\beta$ and $X^\gamma$, the so-called $\beta$- and $\gamma$-duals, also referred to as Köthe–Toeplitz duals by some authors (see [a2]). If $\vert$ denotes any of the symbols $\alpha$, $\beta$ or $\gamma$, then for all $X,Y \subset \omega$ one has: $X \subset X^\Vert = (X^\vert)^\vert$, $X^{\vert\vert\vert} = X^\vert$, and $X \subset Y$ implies $Y^\vert \subset X^\vert$. A set $X \subset \omega$ is called ($\vert$-) perfect if $X = X^\Vert$; $X^\vert$ is perfect, so is $\phi$ (the set of sequences that terminate in zeroes); the set $c$ of convergent sequences is not perfect. For any $X \supset \phi$, $X$ and $X^\alpha$ (and analogously $X$ and $X^\beta$) are in duality with respect to the bilinear functional $(\cdot,\cdot)$ on $X^\alpha \times X$ defined by $(x,y) = \sum_k x_k y_k$, and various topologies may be introduced on $X$ and $X^\alpha$: usually on $X$ the weak topology $\sigma(X,X^\alpha)$, the Mackey topology $\tau(X,X^\alpha)$, or the normal topology is taken. If $X \supset \phi$ and $Y$ are BK-spaces (i.e., Banach FK-spaces), then $Z$ is a BK-space with respect to $\Vert\alpha\Vert = \sup\{\Vert (a_k x_k) \Vert : \Vert x \Vert \le 1 \}$. However, if $X$ is not a BK space, then $Z$ need not even be an FK-space; for instance, $\omega^\beta = \phi$ is not an FK-space. The $\beta$-dual of an FK space $X \supset \phi$ is contained in its continuous dual $X^*$ in the following sense: The mapping $\hat\cdot : X^\beta \rightarrow X^*$ defined by $\hat a = (a,\cdot)$ ($a \in X^\beta$) is linear and one-to-one; if $X \supset \phi$ has the AK-property (i.e. every sequence $x = (x_k) \in X$ has a unique representation $x = \sum_k x_k e^{(k)}$, where for each $k$, $e^{(k)}$ is the sequence with $e^{(k)}_k = 1$ and $e^{(k)}_j = 0$ if $j \ne k$), then $\hat\cdot$ is an isomorphism.

References

[a1]	W.H. Ruckle, "Sequence spaces" , Pitman (1989)
[a2]	A. Wilansky, "Summability through functional analysis" , North-Holland (1984)