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.
is bounded: in this case, the auxiliary normed space is
will be a real or complex vector space (not necessarily a TVS, yet) and
will be a real or complex vector space.
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.
forms a basis of neighborhoods at the origin for a locally convex topological vector space topology
will be a Banach disk in any TVS that contains
forms a Banach space is dependent only on the disk
is a disk in a topological vector space (TVS)
so define the following continuous[5] linear map: If
do not contain any non-trivial vector subspace, which implies that
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.
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
is a bounded Banach disk in a Hausdorff locally convex space
is a convex balanced closed neighborhood of the origin in
is a weakly compact bounded equicontinuous disk in
is a metrizable locally convex TVS then for every bounded subset
is a neighborhood basis at the origin for some locally convex topology
where this value is in fact independent of the representative of the equivalence class
denotes the space induced by a radial disk or the space induced by a bounded disk).
so there is a continuous linear surjective canonical map
where one may verify that the definition does not depend on the representative of the equivalence class
[1] and it has a unique continuous linear canonical extension to
then we may create the auxiliary normed space
then we may create the auxiliary seminormed space
is a weakly closed equicontinuous disk in
by the bipolar theorem, it follows that a continuous linear functional
belongs to the continuous dual space of
A linear map between two TVSs is called infrabounded[5] if it maps Banach disks to bounded disks.