View source for Köthe-Toeplitz dual

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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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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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-spaces (i.e., Banach FK-spaces; cf. [[FK-space|FK-space]]), 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]].

====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>