In functional analysis, a branch of mathematics, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.[1] One method is used if the disk is bounded: in this case, the auxiliary normed space is with norm
Induced by a bounded disk – Banach disks
Throughout this article, will be a real or complex vector space (not necessarily a TVS, yet) and
will be a disk in
Seminormed space induced by a disk
Let will be a real or complex vector space. For any subset
of
the Minkowski functional of
defined by:
- If
then define
to be the trivial map
[2] and it will be assumed that
[note 1]
- If
and if
is absorbing in
then denote the Minkowski functional of
in
by
where for allthis is defined by
Let will be a real or complex vector space. For any subset
of
such that the Minkowski functional
is a seminorm on
let
denote
Assumption (Topology): is endowed with the seminorm topology induced by
which will be denoted by
or
Importantly, this topology stems entirely from the set the algebraic structure of
and the usual topology on
(since
is defined using only the set
and scalar multiplication). This justifies the study of Banach disks and is part of the reason why they play an important role in the theory of nuclear operators and nuclear spaces.
The inclusion map is called the canonical map.[1]
Suppose that is a disk. Then
so that
is absorbing in
the linear span of
The set
of all positive scalar multiples of
forms a basis of neighborhoods at the origin for a locally convex topological vector space topology
on
The Minkowski functional of the disk
in
guarantees that
is well-defined and forms a seminorm on
[3] The locally convex topology induced by this seminorm is the topology
that was defined before.
Banach disk definition
A bounded disk in a topological vector space
such that
is a Banach space is called a Banach disk, infracomplete, or a bounded completant in
If its shown that is a Banach space then
will be a Banach disk in any TVS that contains
as a bounded subset.
This is because the Minkowski functional is defined in purely algebraic terms. Consequently, the question of whether or not
forms a Banach space is dependent only on the disk
and the Minkowski functional
and not on any particular TVS topology that
may carry. Thus the requirement that a Banach disk in a TVS
be a bounded subset of
is the only property that ties a Banach disk's topology to the topology of its containing TVS
Properties of disk induced seminormed spaces
Bounded disks
The following result explains why Banach disks are required to be bounded.
Theorem[4][5][1] — If is a disk in a topological vector space (TVS)
then
is bounded in
if and only if the inclusion map
is continuous.
If the disk is bounded in the TVS
then for all neighborhoods
of the origin in
there exists some
such that
It follows that in this case the topology of
is finer than the subspace topology that
inherits from
which implies that the inclusion map
is continuous. Conversely, if
has a TVS topology such that
is continuous, then for every neighborhood
of the origin in
there exists some
such that
which shows that
is bounded in
Hausdorffness
The space is Hausdorff if and only if
is a norm, which happens if and only if
does not contain any non-trivial vector subspace.[6] In particular, if there exists a Hausdorff TVS topology on
such that
is bounded in
then
is a norm. An example where
is not Hausdorff is obtained by letting
and letting
be the
-axis.
Convergence of nets
Suppose that is a disk in
such that
is Hausdorff and let
be a net in
Then
in
if and only if there exists a net
of real numbers such that
and
for all
; moreover, in this case it will be assumed without loss of generality that
for all
Relationship between disk-induced spaces
If then
and
on
so define the following continuous[5] linear map:
If and
are disks in
with
then call the inclusion map
the canonical inclusion of
into
In particular, the subspace topology that inherits from
is weaker than
's seminorm topology.[5]
The disk as the closed unit ball
The disk is a closed subset of
if and only if
is the closed unit ball of the seminorm
; that is,
If is a disk in a vector space
and if there exists a TVS topology
on
such that
is a closed and bounded subset of
then
is the closed unit ball of
(that is,
) (see footnote for proof).[note 2]
Sufficient conditions for a Banach disk
The following theorem may be used to establish that is a Banach space. Once this is established,
will be a Banach disk in any TVS in which
is bounded.
Theorem[7] — Let be a disk in a vector space
If there exists a Hausdorff TVS topology
on
such that
is a bounded sequentially complete subset of
then
is a Banach space.
Assume without loss of generality that and let
be the Minkowski functional of
Since
is a bounded subset of a Hausdorff TVS,
do not contain any non-trivial vector subspace, which implies that
is a norm. Let
denote the norm topology on
induced by
where since
is a bounded subset of
is finer than
Because is convex and balanced, for any
Let be a Cauchy sequence in
By replacing
with a subsequence, we may assume without loss of generality† that for all
This implies that for any
Since for all
by fixing
and taking the limit (in
) as
it follows that
for each
This implies that
as
which says exactly that
in
This shows that
is complete.
†This assumption is allowed because is a Cauchy sequence in a metric space (so the limits of all subsequences are equal) and a sequence in a metric space converges if and only if every subsequence has a sub-subsequence that converges.
Note that even if is not a bounded and sequentially complete subset of any Hausdorff TVS, one might still be able to conclude that
is a Banach space by applying this theorem to some disk
satisfying
The following are consequences of the above theorem:
- A sequentially complete bounded disk in a Hausdorff TVS is a Banach disk.[5]
- Any disk in a Hausdorff TVS that is complete and bounded (e.g. compact) is a Banach disk.[8]
- The closed unit ball in a Fréchet space is sequentially complete and thus a Banach disk.[5]
Suppose that is a bounded disk in a TVS
- If
is a continuous linear map and
is a Banach disk, then
is a Banach disk and
induces an isometric TVS-isomorphism
Properties of Banach disks
Let be a TVS and let
be a bounded disk in
If is a bounded Banach disk in a Hausdorff locally convex space
and if
is a barrel in
then
absorbs
(that is, there is a number
such that
[4]
If is a convex balanced closed neighborhood of the origin in
then the collection of all neighborhoods
where
ranges over the positive real numbers, induces a topological vector space topology on
When
has this topology, it is denoted by
Since this topology is not necessarily Hausdorff nor complete, the completion of the Hausdorff space
is denoted by
so that
is a complete Hausdorff space and
is a norm on this space making
into a Banach space. The polar of
is a weakly compact bounded equicontinuous disk in
and so is infracomplete.
If is a metrizable locally convex TVS then for every bounded subset
of
there exists a bounded disk
in
such that
and both
and
induce the same subspace topology on
[5]
Induced by a radial disk – quotient
Suppose that is a topological vector space and
is a convex balanced and radial set. Then
is a neighborhood basis at the origin for some locally convex topology
on
This TVS topology
is given by the Minkowski functional formed by
which is a seminorm on
defined by
The topology
is Hausdorff if and only if
is a norm, or equivalently, if and only if
or equivalently, for which it suffices that
be bounded in
The topology
need not be Hausdorff but
is Hausdorff. A norm on
is given by
where this value is in fact independent of the representative of the equivalence class
chosen. The normed space
is denoted by
and its completion is denoted by
If in addition is bounded in
then the seminorm
is a norm so in particular,
In this case, we take
to be the vector space
instead of
so that the notation
is unambiguous (whether
denotes the space induced by a radial disk or the space induced by a bounded disk).[1]
The quotient topology on
(inherited from
's original topology) is finer (in general, strictly finer) than the norm topology.
Canonical maps
The canonical map is the quotient map which is continuous when
has either the norm topology or the quotient topology.[1]
If and
are radial disks such that
then
so there is a continuous linear surjective canonical map
defined by sending
to the equivalence class
where one may verify that the definition does not depend on the representative of the equivalence class
that is chosen.[1] This canonical map has norm
[1] and it has a unique continuous linear canonical extension to
that is denoted by
Suppose that in addition and
are bounded disks in
with
so that
and the inclusion
is a continuous linear map. Let
and
be the canonical maps. Then
and
[1]
Induced by a bounded radial disk
Suppose that is a bounded radial disk. Since
is a bounded disk, if
then we may create the auxiliary normed space
with norm
; since
is radial,
Since
is a radial disk, if
then we may create the auxiliary seminormed space
with the seminorm
; because
is bounded, this seminorm is a norm and
so
Thus, in this case the two auxiliary normed spaces produced by these two different methods result in the same normed space.
Duality
Suppose that is a weakly closed equicontinuous disk in
(this implies that
is weakly compact) and let
Related concepts
A disk in a TVS is called infrabornivorous[5] if it absorbs all Banach disks.
A linear map between two TVSs is called infrabounded[5] if it maps Banach disks to bounded disks.
Fast convergence
A sequence in a TVS
is said to be fast convergent[5] to a point
if there exists a Banach disk
such that both
and the sequence is (eventually) contained in
and
in
Every fast convergent sequence is Mackey convergent.[5]
See also
- Bornological space – Space where bounded operators are continuous
- Injective tensor product
- Locally convex topological vector space – A vector space with a topology defined by convex open sets
- Nuclear operator – Linear operator related to topological vector spaces
- Nuclear space – A generalization of finite-dimensional Euclidean spaces different from Hilbert spaces
- Initial topology – Coarsest topology making certain functions continuous
- Projective tensor product – tensor product defined on two topological vector spaces
- Schwartz topological vector space – topological vector space whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets
- Tensor product of Hilbert spaces – Tensor product space endowed with a special inner product
- Topological tensor product – Tensor product constructions for topological vector spaces
- Ultrabornological space
Notes
References
Bibliography
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