In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, two methods of constructing
normed spaces
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
from
disks were systematically employed by
Alexander Grothendieck to define
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
s and
nuclear space
In mathematics, nuclear spaces are topological vector spaces that can be viewed as a generalization of finite dimensional Euclidean spaces and share many of their desirable properties. Nuclear spaces are however quite different from Hilbert spaces, ...
s.
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
Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular:
*Quotient space (topology), in case of topological spaces
* Quotient space (linear algebra), in case of vector spaces
*Quotient ...
If the disk is both bounded and absorbing then the two auxiliary normed spaces are canonically isomorphic (as
topological vector spaces
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
and as
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "lengt ...
s).
Preliminaries
A subset of a vector space is called a
disk and is said to be disked,
absolutely convex In mathematics, a subset ''C'' of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk.
The disked hull ...
, or convex balanced if it is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and
balanced
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ci ...
.
If
and
are subsets of a vector space
then
absorbs if there exists a real
such that
for any scalar
satisfying
WThe set
is called absorbing in
if
absorbs
for every
A subset
of a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
(TVS)
is said to be
bounded in
if every neighborhood of the origin in
absorbs
A subset of a TVS
is called
bornivorous
In functional analysis, a subset of a real or complex vector space X that has an associated vector bornology \mathcal is called bornivorous and a bornivore if it absorbs every element of \mathcal.
If X is a topological vector space (TVS) then a ...
if it absorbs all bounded subsets of
Induced by a bounded disk – Banach disks
Henceforth,
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
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 K is a subset of a real or complex vector space X, the ...
of
defined by:
*If
then define