Concept

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 (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.

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Official source

https://en.wikipedia.org/wiki/Auxiliary_normed_space

About this result

Related courses (4)

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Related concepts (9)

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In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If is a subset of a real or complex vector space then the or of is defined to be the function valued in the extended real numbers, defined by where the infimum of the empty set is defined to be positive infinity (which is a real number so that would then be real-valued).

Metrizable topological vector space

In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of locally convex metrizable TVS.

Bornivorous set

In functional analysis, a subset of a real or complex vector space that has an associated vector bornology is called bornivorous and a bornivore if it absorbs every element of If is a topological vector space (TVS) then a subset of is bornivorous if it is bornivorous with respect to the von-Neumann bornology of . Bornivorous sets play an important role in the definitions of many classes of topological vector spaces, particularly bornological spaces.