Köthe-Toeplitz dual
For any subset of the set
of all sequences
, the set
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is called a Köthe–Toeplitz or -dual of
. 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
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which for and
, the sets of convergent or bounded series, reduce to
and
, the so-called
- and
-duals, also referred to as Köthe–Toeplitz duals by some authors (see [a2]). If
denotes any of the symbols
,
or
, then for all
one has:
,
, and
implies
. A set
is called (
-) perfect if
;
is perfect, so is
(the set of sequences that terminate in naughts); the set
of convergent sequences is not perfect. For any
,
and
(and analogously
and
) are in duality with respect to the bilinear functional
on
defined by
, and various topologies may be introduced on
and
, usually on
the weak
, the Mackey
, or the normal topology is taken (see [a1]; cf. also Weak topology; Mackey topology; Normal space). If
and
are BK-spaces (i.e., Banach FK-spaces; cf. FK-space), then
is a BK-space with respect to
. However, if
is not a BK space, then
need not even be an FK-space; for instance,
is not an FK-space. The
-dual of an FK space
is contained in its continuous dual
in the following sense: The mapping
defined by
(
) is linear and one-to-one; if
has the AK-property (i.e. every sequence
has a unique representation
, where for each
,
is the sequence with
and
if
), then
is an isomorphism.
References
[a1] | W.H. Ruckle, "Sequence spaces" , Pitman (1989) |
[a2] | A. Wilansky, "Summability through functional analysis" , North-Holland (1984) |
Köthe-Toeplitz dual. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=K%C3%B6the-Toeplitz_dual&oldid=17307