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Concept
Auxiliary normed space
Résumé
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 (as topological vector spaces and as normed spaces).
Throughout this article, will be a real or complex vector space (not necessarily a TVS, yet) and will be a disk in
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 and it will be assumed that
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 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.
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
The locally convex topology induced by this seminorm is the topology that was defined before.
Source officielle
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