In functional analysis, 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

The other method is used if the disk is absorbing: in this case, the auxiliary normed space is the quotient space If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as topological vector spaces and as normed spaces).

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 all this 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

which is called the seminormed space induced by where if is a norm then it is called the normed space induced by

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.

Proof

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.

Proof

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

so that in particular, by taking it follows that is contained in Since is finer than is a Cauchy sequence in For all is a Hausdorff sequentially complete subset of In particular, this is true for so there exists some such that in

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

because

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

be the polar of Because by the bipolar theorem, it follows that a continuous linear functional belongs to if and only if belongs to the continuous dual space of where is the Minkowski functional of defined by [9]

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

Notes

  1. This is the smallest vector space containing Alternatively, if then may instead be replaced with
  2. Assume WLOG that Since is closed in it is also closed in and since the seminorm is the Minkowski functional of which is continuous on it follows Narici & Beckenstein (2011, pp. 119–120) that is the closed unit ball in

References

Bibliography

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