Auxiliary normed space


In functional analysis, two methods of constructing normed spaces from disks were systematically employed by Alexander Grothendieck to define nuclear operators and nuclear spaces.
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.

Preliminaries

Induced by a bounded disk – Banach disks

Seminormed space induced by a disk

Henceforth, will be a real or complex vector space and let will be a disk in.
If then set and give the indiscrete topology.
Henceforth assume that .
Since is a disk, so that is absorbing in, the linear span of.
Note that the set of all positive scalar multiples of forms a basis of neighborhoods at 0 for a locally convex topological vector space topology on.
The Minkowski functional of in, which we denote and define by
is well-defined and forms a seminorm on.
The locally convex topology topology induced by this seminorm is the topology that was defined before.

Banach disk definition

Observe that 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 to the topology of its containing TVS.
;Related definitions

Properties of disk induced seminormed spaces

;Bounded disks
The following result explains why Banach disks are required to be bounded.
;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.
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 we can assume without loss of generality that for all.
;Relationship between disk-induced spaces
Note that if then and on, so we can define the following continuous linear map:
In particular, the subspace topology that inherits from is weaker than 's seminorm topology.
; as the closed unit ball
Clearly, the disk is a closed subset of if and only if is the closed unit ball of the seminorm i.e..

Examples and sufficient conditions

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.
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:
Suppose that is a bounded disk in a TVS.

Properties of Banach disks

Let be a TVS and let be a bounded disk in.

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.
The quotient topology on is 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.
If and are radial disks such that then ⊆ p 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.
This canonical map has norm 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 natural inclusion is a continuous linear map.
Let,, and be the canonical maps.
Then and.

Induced by a bounded radial disk

Suppose that is a bounded radial disk.
Since is a bounded disk, if we let then we may create the auxiliary normed space with norm ; since is radial,.
Since is a radial disk, if we let 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 and let be the polar of.
Since 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 .