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For any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101501.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101502.png" /> of all sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101503.png" />, the set
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For any subset $X$ of the set $\omega$ of all sequences $x = (x_k)$, the set
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$$
 +
X^\alpha = \{ a \in \omega : \sum_k | a_k x_k | < \infty \ \text{for all}\ x \in X \}
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$$
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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|Linear functional]]) and the characterization of [[Matrix|matrix]] transformations between sequence spaces. They are special cases of the more general multiplier sequence spaces
 +
$$
 +
Z = M(X,Y) =  \{ a \in \omega : ax = (a_k x_k) \in Y \ \text{for all}\ x \in X \}
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$$
 +
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 [[#References|[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 space|normal]] topology is taken. If $X \supset \phi$ and $Y$ are [[BK-space]]s (i.e., Banach [[FK-space]]s), 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]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101504.png" /></td> </tr></table>
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====References====
 
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<table>
is called a Köthe–Toeplitz or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101506.png" />-dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101507.png" />. These duals play an important role in the representation of linear functionals (cf. [[Linear functional|Linear functional]]) and the characterization of [[Matrix|matrix]] transformations between sequence spaces. They are special cases of the more general multiplier sequence spaces
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<TR><TD valign="top">[a1]</TD> <TD valign="top"> W.H. Ruckle,  "Sequence spaces" , Pitman  (1989)</TD></TR>
 
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<TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Wilansky,  "Summability through functional analysis" , North-Holland  (1984)</TD></TR>
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101508.png" /></td> </tr></table>
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</table>
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k1101509.png" /></td> </tr></table>
 
  
which for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015011.png" />, the sets of convergent or bounded series, reduce to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015013.png" />, the so-called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015015.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015017.png" />-duals, also referred to as Köthe–Toeplitz duals by some authors (see [[#References|[a2]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015018.png" /> denotes any of the symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015020.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015021.png" />, then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015022.png" /> one has: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015025.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015026.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015027.png" /> is called (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015031.png" />-) perfect if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015032.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015033.png" /> is perfect, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015034.png" /> (the set of sequences that terminate in naughts); the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015035.png" /> of convergent sequences is not perfect. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015038.png" /> (and analogously <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015040.png" />) are in [[Duality|duality]] with respect to the [[Bilinear functional|bilinear functional]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015041.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015042.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015043.png" />, and various topologies may be introduced on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015045.png" />, usually on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015046.png" /> the weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015047.png" />, the Mackey <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015048.png" />, or the normal topology is taken (see [[#References|[a1]]]; cf. also [[Weak topology|Weak topology]]; [[Mackey topology|Mackey topology]]; [[Normal space|Normal space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015050.png" /> are [[BK-space]]s (i.e., Banach [[FK-space]]s), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015051.png" /> is a BK-space with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015052.png" />. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015053.png" /> is not a BK space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015054.png" /> need not even be an [[FK-space|FK-space]]; for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015055.png" /> is not an FK-space. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015056.png" />-dual of an FK space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015057.png" /> is contained in its continuous dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015058.png" /> in the following sense: The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015059.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015060.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015061.png" />) is linear and one-to-one; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015062.png" /> has the AK-property (i.e. every sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015063.png" /> has a unique representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015064.png" />, where for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015066.png" /> is the sequence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015068.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015069.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k110/k110150/k11015070.png" /> is an [[Isomorphism|isomorphism]].
+
{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.H. Ruckle,  "Sequence spaces" , Pitman  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Wilansky,  "Summability through functional analysis" , North-Holland  (1984)</TD></TR></table>
 

Latest revision as of 17:02, 4 October 2017

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)
How to Cite This Entry:
Köthe-Toeplitz dual. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%B6the-Toeplitz_dual&oldid=41997
This article was adapted from an original article by E. Malkowsky (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article