In mathematics, a dual system, dual pair or a duality over a field
is a triple
consisting of two vector spaces
and
over
and a non-degenerate bilinear map
.
Mathematical duality, the study of dual systems, has an important place in functional analysis. It has extensive applications to quantum mechanics arising in the theory of Hilbert spaces.
Definition, notation, and conventions
Pairings
A pairing or pair over a field
is a triple
which may also be denoted by
consisting of two vector spaces
and
over
and a bilinear map
, named the bilinear map associated with the pairing, more simply the pairing's map or its bilinear form. The examples here only describe
which is either the real numbers
or the complex numbers
.
For every
, define
![{\displaystyle {\begin{alignedat}{4}b(x,\,\cdot \,):\,&Y&&\to &&\,\mathbb {K} \\&y&&\mapsto &&\,b(x,y)\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1225515ecc2edd7072c6ba38c206d54f60fc5ec)
and for every
![{\displaystyle y\in Y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e1353f0febe9bcf693d849ec82ce8d94e5f7bc)
define
![{\displaystyle {\begin{alignedat}{4}b(\,\cdot \,,y):\,&X&&\to &&\,\mathbb {K} \\&x&&\mapsto &&\,b(x,y).\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cba46864a895c5d7fe06a4fe183c03157c38a31a)
Every
![{\displaystyle b(x,\,\cdot \,)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5293773749dbc11221eaab0d4c913ac6ecf27fc8)
is a linear functional on
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
and every
![{\displaystyle b(\,\cdot \,,y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a3d7d444875a36f4c20a36aa430d6b2f085f0c)
is a linear functional on
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
Let
![{\displaystyle b(X,\,\cdot \,):=\{b(x,\,\cdot \,):x\in X\}\qquad {\text{ and }}\qquad b(\,\cdot \,,Y):=\{b(\,\cdot \,,y):y\in Y\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc2af769af56de0754ee318b20514cd43c52331)
Where each of these sets forms a vector space of linear functionals.
It is common practice to write
instead of
, in which case the pairing may often be denoted by
rather than
. However, this article will reserve the use of
for the canonical evaluation map (defined below) so as to avoid confusion for readers not familiar with this subject.
Dual pairings
A pairing
is called a dual system, a dual pair, or a duality over
if the bilinear form
is non-degenerate, which means that it satisfies the following two separation axioms:
separates (distinguishes) points of
: if
is such that
then
; or equivalently, for all non-zero
, the map
is not identically
(i.e. there exists a
such that
for each
);
separates (distinguishes) points of
: if
is such that
then
; or equivalently, for all non-zero
the map
is not identically
(i.e. there exists an
such that
for each
).
In this case
is non-degenerate, and one can say that
places
and
in duality (or, redundantly but explicitly, in separated duality), and
is called the duality pairing of the triple
.
Total subsets
A subset
of
is called total if for every
,
![{\displaystyle b(x,s)=0\quad {\text{ for all }}s\in S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ccf0704ba82200cb30f6c36b387d64a6a57891)
implies
![{\displaystyle x=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71320d636c7a43546bf4e94edb94649c5b2e82b9)
A total subset of
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is defined analogously (see footnote).
[note 1] Thus
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
separates points of
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
if and only if
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is a total subset of
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
, and similarly for
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
.
Orthogonality
The vectors
and
are called orthogonal, written
, if
. Two subsets
and
are orthogonal, written
, if
; that is, if
for all
and
. The definition of a subset being orthogonal to a vector is defined analogously.
The orthogonal complement or annihilator of a subset
is
![{\displaystyle R^{\perp }:=\{y\in Y:R\perp y\}:=\{y\in Y:b(R,y)=\{0\}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb7840e0c7fd7d95d8d2dc842e098734ce08623)
Thus
![{\displaystyle R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33)
is a total subset of
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
if and only if
![{\displaystyle R^{\perp }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9784d45eaf020dc36f7b8e54633ac74869389cbd)
equals
![{\displaystyle \{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694)
.
Polar sets
Given a triple
defining a pairing over
, the absolute polar set or polar set of a subset
of
is the set:
![{\displaystyle A^{\circ }:=\left\{y\in Y:\sup _{x\in A}|b(x,y)|\leq 1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68c63e2282b6d38b1e152a9fea5435999e613680)
Symmetrically, the absolute polar set or polar set of a subset
![{\displaystyle B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a)
of
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is denoted by
![{\displaystyle B^{\circ }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f22983da18c7ba4ec07b7af3514660752bd2e2a)
and defined by
![{\displaystyle B^{\circ }:=\left\{x\in X:\sup _{y\in B}|b(x,y)|\leq 1\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c49953d293b1c140d273e50ecef34eb24aa8561d)
To use bookkeeping that helps keep track of the anti-symmetry of the two sides of the duality, the absolute polar of a subset
of
may also be called the absolute prepolar or prepolar of
and then may be denoted by
The polar
is necessarily a convex set containing
where if
is balanced then so is
and if
is a vector subspace of
then so too is
a vector subspace of
If
is a vector subspace of
then
and this is also equal to the real polar of
If
then the bipolar of
, denoted
, is the polar of the orthogonal complement of
, i.e., the set
Similarly, if
then the bipolar of
is
Dual definitions and results
Given a pairing
define a new pairing
where
for all
and
.
There is a consistent theme in duality theory that any definition for a pairing
has a corresponding dual definition for the pairing
- Convention and Definition: Given any definition for a pairing
one obtains a dual definition by applying it to the pairing
This conventions also apply to theorems.
For instance, if "
distinguishes points of
" (resp, "
is a total subset of
") is defined as above, then this convention immediately produces the dual definition of "
distinguishes points of
" (resp, "
is a total subset of
").
This following notation is almost ubiquitous and allows us to avoid assigning a symbol to
- Convention and Notation: If a definition and its notation for a pairing
depends on the order of
and
(for example, the definition of the Mackey topology
on
) then by switching the order of
and
then it is meant that definition applied to
(continuing the same example, the topology
would actually denote the topology
).
For another example, once the weak topology on
is defined, denoted by
, then this dual definition would automatically be applied to the pairing
so as to obtain the definition of the weak topology on
, and this topology would be denoted by
rather than
.
Identification of
with
Although it is technically incorrect and an abuse of notation, this article will adhere to the nearly ubiquitous convention of treating a pairing
interchangeably with
and also of denoting
by
Examples
Restriction of a pairing
Suppose that
is a pairing,
is a vector subspace of
and
is a vector subspace of
. Then the restriction of
to
is the pairing
If
is a duality, then it's possible for a restriction to fail to be a duality (e.g. if
and
).
This article will use the common practice of denoting the restriction
by
Canonical duality on a vector space
Suppose that
is a vector space and let
denote the algebraic dual space of
(that is, the space of all linear functionals on
). There is a canonical duality
where
which is called the evaluation map or the natural or canonical bilinear functional on
Note in particular that for any
is just another way of denoting
; i.e.
If
is a vector subspace of
, then the restriction of
to
is called the canonical pairing where if this pairing is a duality then it is instead called the canonical duality. Clearly,
always distinguishes points of
, so the canonical pairing is a dual system if and only if
separates points of
The following notation is now nearly ubiquitous in duality theory.
The evaluation map will be denoted by
(rather than by
) and
will be written rather than
- Assumption: As is common practice, if
is a vector space and
is a vector space of linear functionals on
then unless stated otherwise, it will be assumed that they are associated with the canonical pairing ![{\displaystyle \langle X,N\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41b94196a783eec873e450f5c142f58adf979a41)
If
is a vector subspace of
then
distinguishes points of
(or equivalently,
is a duality) if and only if
distinguishes points of
or equivalently if
is total (that is,
for all
implies
).
Canonical duality on a topological vector space
Suppose
is a topological vector space (TVS) with continuous dual space
Then the restriction of the canonical duality
to
×
defines a pairing
for which
separates points of
If
separates points of
(which is true if, for instance,
is a Hausdorff locally convex space) then this pairing forms a duality.
- Assumption: As is commonly done, whenever
is a TVS, then unless indicated otherwise, it will be assumed without comment that it's associated with the canonical pairing ![{\displaystyle \left\langle X,X^{\prime }\right\rangle .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fec39244aa3bffa99a35dc701dafebd8e7dd7a6a)
Polars and duals of TVSs
The following result shows that the continuous linear functionals on a TVS are exactly those linear functionals that are bounded on a neighborhood of the origin.
Inner product spaces and complex conjugate spaces
A pre-Hilbert space
is a dual pairing if and only if
is vector space over
or
has dimension
Here it is assumed that the sesquilinear form
is conjugate homogeneous in its second coordinate and homogeneous in its first coordinate.
- If
is a real Hilbert space then
forms a dual system. - If
is a complex Hilbert space then
forms a dual system if and only if
If
is non-trivial then
does not even form pairing since the inner product is sesquilinear rather than bilinear.
Suppose that
is a complex pre-Hilbert space with scalar multiplication denoted as usual by juxtaposition or by a dot
Define the map
![{\displaystyle \,\cdot \,\perp \,\cdot \,:\mathbb {C} \times H\to H\quad {\text{ by }}\quad c\perp x:={\overline {c}}x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6305d6c4142ea669e7aceb06d581a4ac2a79f09d)
where the right-hand side uses the scalar multiplication of
![{\displaystyle H.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8933ae7244305ae7824aa18e077d1cf946e2ee9d)
Let
![{\displaystyle {\overline {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4)
denote the
complex conjugate vector space of
![{\displaystyle H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704)
where
![{\displaystyle {\overline {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4)
denotes the additive group of
![{\displaystyle (H,+)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b5e04d98957dae89ad5e4aed4c9c1f4e17a7af)
(so vector addition in
![{\displaystyle {\overline {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4)
is identical to vector addition in
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
) but with scalar multiplication in
![{\displaystyle {\overline {H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e3c5eac166e464406970ed2cadc14fa7345da4)
being the map
![{\displaystyle \,\cdot \,\perp \,\cdot \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1a3bdbac1bf7b17120055cbb80900975e0ae501)
(instead of the scalar multiplication that
![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
is endowed with).
The map
defined by
is linear in both coordinates[note 2] and so
forms a dual pairing.
Other examples
- Suppose
and for all
let ![{\displaystyle b\left(\left(x_{1},y_{1}\right),\left(x_{2},y_{2},z_{2}\right)\right):=x_{1}x_{2}+y_{1}y_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2434a1005b241549110de523092fd56e5fef7303)
Then
is a pairing such that
distinguishes points of
but
does not distinguish points of
Furthermore, ![{\displaystyle X^{\perp }:=\{y\in Y:X\perp y\}=\{(0,0,z):z\in \mathbb {R} \}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/154287a8ea325d7707c3d922cc2835cf86687525)
- Let
(where
is such that
), and
Then
is a dual system. - Let
and
be vector spaces over the same field
Then the bilinear form
places
and
in duality. - A sequence space
and its beta dual
with the bilinear map defined as
for
forms a dual system.
Weak topology
Suppose that
is a pairing of vector spaces over
If
then the weak topology on
induced by
(and
) is the weakest TVS topology on
denoted by
or simply
making all maps
continuous as
ranges over
If
is not clear from context then it should be assumed to be all of
in which case it is called the weak topology on
(induced by
). The notation
or (if no confusion could arise) simply
is used to denote
endowed with the weak topology
Importantly, the weak topology depends entirely on the function
the usual topology on
and
's vector space structure but not on the algebraic structures of
Similarly, if
then the dual definition of the weak topology on
induced by
(and
), which is denoted by
or simply
(see footnote for details).[note 3]
- Definition and Notation: If "
" is attached to a topological definition (e.g.
-converges,
-bounded,
etc.) then it means that definition when the first space (i.e.
) carries the
topology. Mention of
or even
and
may be omitted if no confusion arises. So, for instance, if a sequence
in
"
-converges" or "weakly converges" then this means that it converges in
whereas if it were a sequence in
, then this would mean that it converges in
).
The topology
is locally convex since it is determined by the family of seminorms
defined by
as
ranges over
If
and
is a net in
then
-converges to
if
converges to
in
A net
-converges to
if and only if for all
converges to
If
is a sequence of orthonormal vectors in Hilbert space, then
converges weakly to 0 but does not norm-converge to 0 (or any other vector).
If
is a pairing and
is a proper vector subspace of
such that
is a dual pair, then
is strictly coarser than
Bounded subsets
A subset
of
is
-bounded if and only if
![{\displaystyle \sup _{}|b(S,y)|<\infty \quad {\text{ for all }}y\in Y,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bdd93b0165602359c6f9ce41b8cf05341c9c28a2)
where
Hausdorffness
If
is a pairing then the following are equivalent:
distinguishes points of
; - The map
defines an injection from
into the algebraic dual space of
;
is Hausdorff.
Weak representation theorem
The following theorem is of fundamental importance to duality theory because it completely characterizes the continuous dual space of
Weak representation theorem — Let
be a pairing over the field
Then the continuous dual space of
is
![{\displaystyle b(\,\cdot \,,Y):=\{b(\,\cdot \,,y):y\in Y\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f12223d8c73de7d5529a7cc03d04acc1eaef200)
Furthermore,
- If
is a continuous linear functional on
then there exists some
such that
; if such a
exists then it is unique if and only if
distinguishes points of
- Note that whether or not
distinguishes points of
is not dependent on the particular choice of ![{\displaystyle y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f)
- The continuous dual space of
may be identified with the quotient space
where
- This is true regardless of whether or not
distinguishes points of
or
distinguishes points of ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
Consequently, the continuous dual space of
is
![{\displaystyle (X,\sigma (X,Y,b))^{\prime }=b(\,\cdot \,,Y):=\left\{b(\,\cdot \,,y):y\in Y\right\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f08135ea3133403f5bd290859de6b5ae9b016b3)
With respect to the canonical pairing, if
is a TVS whose continuous dual space
separates points on
(i.e. such that
is Hausdorff, which implies that
is also necessarily Hausdorff) then the continuous dual space of
is equal to the set of all "evaluation at a point
" maps as
ranges over
(i.e. the map that send
to
). This is commonly written as
![{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right)^{\prime }=X\qquad {\text{ or }}\qquad \left(X_{\sigma }^{\prime }\right)^{\prime }=X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b45f4532d152c2c5fe36380bd72e219396a0bebd)
This very important fact is why results for polar topologies on continuous dual spaces, such as the
strong dual topology ![{\displaystyle \beta \left(X^{\prime },X\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23a12a5497302faea7ad4a65cf78728e273a7a0d)
on
![{\displaystyle X^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
for example, can also often be applied to the original TVS
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
; for instance,
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
being identified with
![{\displaystyle \left(X_{\sigma }^{\prime }\right)^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1f19485c73fc068ce725ee7127930492d26623)
means that the topology
![{\displaystyle \beta \left(\left(X_{\sigma }^{\prime }\right)^{\prime },X_{\sigma }^{\prime }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ebfcf62eec2c54061f04e3118043884b61c16f)
on
![{\displaystyle \left(X_{\sigma }^{\prime }\right)^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1f19485c73fc068ce725ee7127930492d26623)
can instead be thought of as a topology on
![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
Moreover, if
![{\displaystyle X^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
is endowed with a topology that is
finer than
![{\displaystyle \sigma \left(X^{\prime },X\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43ed437c13fa2a58f9603401ef4229ca41d82a03)
then the continuous dual space of
![{\displaystyle X^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
will necessarily contain
![{\displaystyle \left(X_{\sigma }^{\prime }\right)^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee1f19485c73fc068ce725ee7127930492d26623)
as a subset. So for instance, when
![{\displaystyle X^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a3a5819cc45f097de14b3ac5a8bedd902bc66d)
is endowed with the strong dual topology (and so is denoted by
![{\displaystyle X_{\beta }^{\prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9afb4be9e601adc17a887dc64d9bd792c282d3a)
) then
![{\displaystyle \left(X_{\beta }^{\prime }\right)^{\prime }~\supseteq ~\left(X_{\sigma }^{\prime }\right)^{\prime }~=~X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f78b6c639d0420dd9a815adf5d3e261b88ef99f7)
which (among other things) allows for
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
to be endowed with the subspace topology induced on it by, say, the strong dual topology
![{\displaystyle \beta \left(\left(X_{\beta }^{\prime }\right)^{\prime },X_{\beta }^{\prime }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba97acefb20055e6f563e1a851d9039e2ffe148a)
(this topology is also called the strong
bidual topology and it appears in the theory of
reflexive spaces: the Hausdorff locally convex TVS
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is said to be
semi-reflexive if
![{\displaystyle \left(X_{\beta }^{\prime }\right)^{\prime }=X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83104343f969bc166e2f3891a529c455741014eb)
and it will be called
reflexive if in addition the strong bidual topology
![{\displaystyle \beta \left(\left(X_{\beta }^{\prime }\right)^{\prime },X_{\beta }^{\prime }\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba97acefb20055e6f563e1a851d9039e2ffe148a)
on
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
is equal to
![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab)
's original/starting topology).
Orthogonals, quotients, and subspaces
If
is a pairing then for any subset
of
:
and this set is
-closed;
; - Thus if
is a
-closed vector subspace of
then ![{\displaystyle S\subseteq S^{\perp \perp }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef0edf78b4e438212057510e37bbb766bc003df)
- If
is a family of
-closed vector subspaces of
then ![{\displaystyle \left(\bigcap _{i\in I}S_{i}\right)^{\perp }=\operatorname {cl} _{\sigma (Y,X,b)}\left(\operatorname {span} \left(\bigcup _{i\in I}S_{i}^{\perp }\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea06736df344bf1f1eaf4865a586e07ac77b4718)
- If
is a family of subsets of
then ![{\displaystyle \left(\bigcup _{i\in I}S_{i}\right)^{\perp }=\bigcap _{i\in I}S_{i}^{\perp }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f425731fa2c332546b64b7ad359b967467f32ef)
If
is a normed space then under the canonical duality,
is norm closed in
and
is norm closed in
Subspaces
Suppose that
is a vector subspace of
and let
denote the restriction of
to
The weak topology
on
is identical to the subspace topology that
inherits from
Also,
is a paired space (where
means
) where
is defined by
![{\displaystyle \left(m,y+M^{\perp }\right)\mapsto b(m,y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81f288d9c649f0bb30bab4aa91fa3fa4fbe5a3c0)
The topology
is equal to the subspace topology that
inherits from
Furthermore, if
is a dual system then so is
Quotients
Suppose that
is a vector subspace of
Then
is a paired space where
is defined by
![{\displaystyle (x+M,y)\mapsto b(x,y).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c760923f2533dd43d2a2eda2adcd24aedb5389)
The topology
is identical to the usual quotient topology induced by
on
Polars and the weak topology
If
is a locally convex space and if
is a subset of the continuous dual space
then
is
-bounded if and only if
for some barrel
in
The following results are important for defining polar topologies.
If
is a pairing and
then:
- The polar
of
is a closed subset of ![{\displaystyle (Y,\sigma (Y,X,b)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/458520bf750af18ed32987f5c1d68500617a9813)
- The polars of the following sets are identical: (a)
; (b) the convex hull of
; (c) the balanced hull of
; (d) the
-closure of
; (e) the
-closure of the convex balanced hull of ![{\displaystyle A.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a71bf21ad35b8fe05555041d54d1e17eeb0f490)
- The bipolar theorem: The bipolar of
denoted by
is equal to the
-closure of the convex balanced hull of
- The bipolar theorem in particular "is an indispensable tool in working with dualities."
is
-bounded if and only if
is absorbing in ![{\displaystyle Y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c668649af47a30006f93c9847d61fee8d9ffb61)
- If in addition
distinguishes points of
then
is
-bounded if and only if it is
-totally bounded.
If
is a pairing and
is a locally convex topology on
that is consistent with duality, then a subset
of
is a barrel in
if and only if
is the polar of some
-bounded subset of
Transposes
Transposes of a linear map with respect to pairings
Let
and
be pairings over
and let
be a linear map.
For all
let
be the map defined by
It is said that
's transpose or adjoint is well-defined if the following conditions are satisfied:
distinguishes points of
(or equivalently, the map
from
into the algebraic dual
is injective), and
where
and
.
In this case, for any
there exists (by condition 2) a unique (by condition 1)
such that
), where this element of
will be denoted by
This defines a linear map
![{\displaystyle {}^{t}F:Z\to Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b2001ad80b4d0e260e83b50f3e41ad30414cdc1)
called the transpose or adjoint of
with respect to
and
(this should not be confused with the Hermitian adjoint). It is easy to see that the two conditions mentioned above (i.e. for "the transpose is well-defined") are also necessary for
to be well-defined. For every
the defining condition for
is
![{\displaystyle c(F(\,\cdot \,),z)=b\left(\,\cdot \,,{}^{t}F(z)\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e000ad4fd0b092e1d5dbe8e6931a2ce6662bda)
that is,
![{\displaystyle c(F(x),z)=b\left(x,{}^{t}F(z)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3065fc7c298fb0207b0258386b040f0cabc1af)
for all
By the conventions mentioned at the beginning of this article, this also defines the transpose of linear maps of the form
[note 4]
[note 5]
[note 6]
[note 7] etc. (see footnote).
Properties of the transpose
Throughout,
and
be pairings over
and
will be a linear map whose transpose
is well-defined.
is injective (i.e.
) if and only if the range of
is dense in ![{\displaystyle \left(W,\sigma \left(W,Z,c\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5753db1d7635de2862a4feb680dbba0725a6eff8)
- If in addition to
being well-defined, the transpose of
is also well-defined then ![{\displaystyle {}^{tt}F=F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8107d2bd20d375c0633be02d891a74107574595)
- Suppose
is a pairing over
and
is a linear map whose transpose
is well-defined. Then the transpose of
which is
is well-defined and ![{\displaystyle {}^{t}(F\circ E)={}^{t}E\circ {}^{t}F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fee4b37c181abfc1fba6354e33a9187cac2ba22d)
- If
is a vector space isomorphism then
is bijective, the transpose of
which is
is well-defined, and ![{\displaystyle {}^{t}\left(F^{-1}\right)=\left({}^{t}F\right)^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed35a59a535f8d6ce37b4518032a92b36ced8569)
- Let
and let
denotes the absolute polar of
then:
; - if
for some
then
; - if
is such that
then
; - if
and
are weakly closed disks then
if and only if
; ![{\displaystyle \operatorname {ker} {}^{t}F=[F(X)]^{\perp }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b541687345423da736a58a0e4fe8f6ef17c72f8)
- These results hold when the real polar is used in place of the absolute polar.
If
and
are normed spaces under their canonical dualities and if
is a continuous linear map, then
Weak continuity
A linear map
is weakly continuous (with respect to
and
) if
is continuous.
The following result shows that the existence of the transpose map is intimately tied to the weak topology.
Proposition — Assume that
distinguishes points of
and
is a linear map. Then the following are equivalent:
is weakly continuous (that is,
is continuous);
; - the transpose of
is well-defined.
If
is weakly continuous then
is weakly continuous, meaning that
is continuous; - the transpose of
is well-defined if and only if
distinguishes points of
in which case ![{\displaystyle {}^{tt}F=F.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8107d2bd20d375c0633be02d891a74107574595)
Weak topology and the canonical duality
Suppose that
is a vector space and that
is its the algebraic dual. Then every
-bounded subset of
is contained in a finite dimensional vector subspace and every vector subspace of
is
-closed.
Weak completeness
If
is a complete topological vector space say that
is
-complete or (if no ambiguity can arise) weakly-complete. There exist Banach spaces that are not weakly-complete (despite being complete in their norm topology).
If
is a vector space then under the canonical duality,
is complete. Conversely, if
is a Hausdorff locally convex TVS with continuous dual space
then
is complete if and only if
; that is, if and only if the map
defined by sending
to the evaluation map at
(i.e.
) is a bijection.
In particular, with respect to the canonical duality, if
is a vector subspace of
such that
separates points of
then
is complete if and only if
Said differently, there does not exist a proper vector subspace
of
such that
is Hausdorff and
is complete in the weak-* topology (i.e. the topology of pointwise convergence). Consequently, when the continuous dual space
of a Hausdorff locally convex TVS
is endowed with the weak-* topology, then
is complete if and only if
(that is, if and only if every linear functional on
is continuous).
Identification of Y with a subspace of the algebraic dual
If
distinguishes points of
and if
denotes the range of the injection
then
is a vector subspace of the algebraic dual space of
and the pairing
becomes canonically identified with the canonical pairing
(where
is the natural evaluation map). In particular, in this situation it will be assumed without loss of generality that
is a vector subspace of
's algebraic dual and
is the evaluation map.
- Convention: Often, whenever
is injective (especially when
forms a dual pair) then it is common practice to assume without loss of generality that
is a vector subspace of the algebraic dual space of
that
is the natural evaluation map, and also denote
by ![{\displaystyle X^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/980210d5ccf78c678264901dc7b0ce8a53d827bc)
In a completely analogous manner, if
distinguishes points of
then it is possible for
to be identified as a vector subspace of
's algebraic dual space.
Algebraic adjoint
In the special case where the dualities are the canonical dualities
and
the transpose of a linear map
is always well-defined. This transpose is called the algebraic adjoint of
and it will be denoted by
; that is,
In this case, for all
where the defining condition for
is:
![{\displaystyle \left\langle x,F^{\#}\left(w^{\prime }\right)\right\rangle =\left\langle F(x),w^{\prime }\right\rangle \quad {\text{ for all }}>x\in X,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f714c257ae5012436e2db6ef24386d5ef5f3ee1d)
or equivalently,
If
for some integer
is a basis for
with dual basis
is a linear operator, and the matrix representation of
with respect to
is
then the transpose of
is the matrix representation with respect to
of
Weak continuity and openness
Suppose that
and
are canonical pairings (so
and
) that are dual systems and let
be a linear map. Then
is weakly continuous if and only if it satisfies any of the following equivalent conditions:
is continuous; ![{\displaystyle F^{\#}(Z)\subseteq Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/919fd70270a2122634161c6f27267a34d4aad3d4)
- the transpose of F,
with respect to
and
is well-defined.
If
is weakly continuous then
will be continuous and furthermore,
A map
between topological spaces is relatively open if
is an open mapping, where
is the range of
Suppose that
and
are dual systems and
is a weakly continuous linear map. Then the following are equivalent:
is relatively open; - The range of
is
-closed in
; ![{\displaystyle \operatorname {Im} {}^{t}F=(\operatorname {ker} F)^{\perp }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba6319e2be52a9de3f0a36146e050a5bb19e56a)
Furthermore,
is injective (resp. bijective) if and only if
is surjective (resp. bijective);
is surjective if and only if
is relatively open and injective.
Transpose of a map between TVSs
The transpose of map between two TVSs is defined if and only if
is weakly continuous.
If
is a linear map between two Hausdorff locally convex topological vector spaces then:
- If
is continuous then it is weakly continuous and
is both Mackey continuous and strongly continuous. - If
is weakly continuous then it is both Mackey continuous and strongly continuous (defined below). - If
is weakly continuous then it is continuous if and only if
maps equicontinuous subsets of
to equicontinuous subsets of ![{\displaystyle X^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/980210d5ccf78c678264901dc7b0ce8a53d827bc)
- If
and
are normed spaces then
is continuous if and only if it is weakly continuous, in which case ![{\displaystyle \|F\|=\left\|{}^{t}F\right\|.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f66164141a1352c7ae77410d69a4461a4f11f11)
- If
is continuous then
is relatively open if and only if
is weakly relatively open (i.e.
is relatively open) and every equicontinuous subsets of
is the image of some equicontinuous subsets of ![{\displaystyle Y^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96afe87de56921cbdd12fdca82507cdd1d5d3340)
- If
is continuous injection then
is a TVS-embedding (or equivalently, a topological embedding) if and only if every equicontinuous subsets of
is the image of some equicontinuous subsets of ![{\displaystyle Y^{\prime }.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96afe87de56921cbdd12fdca82507cdd1d5d3340)
Metrizability and separability
Let
be a locally convex space with continuous dual space
and let
- If
is equicontinuous or
-compact, and if
is such that
is dense in
then the subspace topology that
inherits from
is identical to the subspace topology that
inherits from ![{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2781ad2b9271a1990645b00490aa412d366c77b3)
- If
is separable and
is equicontinuous then
when endowed with the subspace topology induced by
is metrizable. - If
is separable and metrizable, then
is separable. - If
is a normed space then
is separable if and only if the closed unit call the continuous dual space of
is metrizable when given the subspace topology induced by ![{\displaystyle \left(X^{\prime },\sigma \left(X^{\prime },X\right)\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2781ad2b9271a1990645b00490aa412d366c77b3)
- If
is a normed space whose continuous dual space is separable (when given the usual norm topology), then
is separable.
Polar topologies and topologies compatible with pairing
Starting with only the weak topology, the use of polar sets produces a range of locally convex topologies. Such topologies are called polar topologies. The weak topology is the weakest topology of this range.
Throughout,
will be a pairing over
and
will be a non-empty collection of
-bounded subsets of
Polar topologies
Given a collection
of subsets of
, the polar topology on
determined by
(and
) or the
-topology on
is the unique topological vector space (TVS) topology on
for which
![{\displaystyle \left\{rG^{\circ }:G\in {\mathcal {G}},r>0\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2901d037ea35630147eebee6c4a1000eb6b9567)
forms a
subbasis of neighborhoods at the origin. When
![{\displaystyle Y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f)
is endowed with this
![{\displaystyle {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
-topology then it is denoted by
Y![{\displaystyle {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
. Every polar topology is necessarily
locally convex. When
![{\displaystyle {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a980c59d42c003fd07fdf3646e1fb95ff82f99)
is a
directed set with respect to subset inclusion (i.e. if for all
![{\displaystyle G,K\in {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42c5e9d8e1215d054c5f3ea7ccdf9a541d1dd708)
there exists some
![{\displaystyle K\in {\mathcal {G}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b522d065c834947a951aefa38f67a7447d448e)
such that
![{\displaystyle G\cup H\subseteq K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab954e674e78b51d7f49c1d9faaf5d58cabc7bc5)
) then this neighborhood subbasis at 0 actually forms a
neighborhood basis at 0.
The following table lists some of the more important polar topologies.
- Notation: If
denotes a polar topology on
then
endowed with this topology will be denoted by
or simply
(e.g. for
we'd have
so that
and
all denote
endowed with
).
("topology of uniform convergence on ...") | Notation | Name ("topology of...") | Alternative name |
finite subsets of ![{\displaystyle X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab) (or -closed disked hulls of finite subsets of ) | ![{\displaystyle \sigma (X,Y,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad3b8a1188a203e79c49539eff3cb417c9720a5)
| pointwise/simple convergence | weak/weak* topology |
-compact disks | | | Mackey topology |
-compact convex subsets | | compact convex convergence | |
-compact subsets (or balanced -compact subsets) | | compact convergence | |
-bounded subsets | ![{\displaystyle b(X,Y,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/707033176b61470aa238662bb9ea78887db4d18f)
| bounded convergence | strong topology Strongest polar topology |
Definitions involving polar topologies
Continuity
A linear map
is Mackey continuous (with respect to
and
) if
is continuous.
A linear map
is strongly continuous (with respect to
and
) if
is continuous.
Bounded subsets
A subset of
is weakly bounded (resp. Mackey bounded, strongly bounded) if it is bounded in
(resp. bounded in
bounded in
).
Topologies compatible with a pair
If
is a pairing over
and
is a vector topology on
then
is a topology of the pairing and that it is compatible (or consistent) with the pairing
if it is locally convex and if the continuous dual space of
[note 8] If
distinguishes points of
then by identifying
as a vector subspace of
's algebraic dual, the defining condition becomes:
Some authors (e.g. [Trèves 2006] and [Schaefer 1999]) require that a topology of a pair also be Hausdorff, which it would have to be if
distinguishes the points of
(which these authors assume).
The weak topology
is compatible with the pairing
(as was shown in the Weak representation theorem) and it is in fact the weakest such topology. There is a strongest topology compatible with this pairing and that is the Mackey topology. If
is a normed space that is not reflexive then the usual norm topology on its continuous dual space is not compatible with the duality
Mackey–Arens theorem
The following is one of the most important theorems in duality theory.
Mackey–Arens theorem I — Let
will be a pairing such that
distinguishes the points of
and let
be a locally convex topology on
(not necessarily Hausdorff). Then
is compatible with the pairing
if and only if
is a polar topology determined by some collection
of
-compact disks that cover[note 9]
It follows that the Mackey topology
which recall is the polar topology generated by all
-compact disks in
is the strongest locally convex topology on
that is compatible with the pairing
A locally convex space whose given topology is identical to the Mackey topology is called a Mackey space. The following consequence of the above Mackey-Arens theorem is also called the Mackey-Arens theorem.
Mackey–Arens theorem II — Let
will be a pairing such that
distinguishes the points of
and let
be a locally convex topology on
Then
is compatible with the pairing if and only if
Mackey's theorem, barrels, and closed convex sets
If
is a TVS (over
or
) then a half-space is a set of the form
for some real
and some continuous real linear functional
on
The above theorem implies that the closed and convex subsets of a locally convex space depend entirely on the continuous dual space. Consequently, the closed and convex subsets are the same in any topology compatible with duality;that is, if
and
are any locally convex topologies on
with the same continuous dual spaces, then a convex subset of
is closed in the
topology if and only if it is closed in the
topology. This implies that the
-closure of any convex subset of
is equal to its
-closure and that for any
-closed disk
in
In particular, if
is a subset of
then
is a barrel in
if and only if it is a barrel in
The following theorem shows that barrels (i.e. closed absorbing disks) are exactly the polars of weakly bounded subsets.
If
is a topological vector space then:
- A closed absorbing and balanced subset
of
absorbs each convex compact subset of
(i.e. there exists a real
such that
contains that set). - If
is Hausdorff and locally convex then every barrel in
absorbs every convex bounded complete subset of ![{\displaystyle X.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba76c5a460c4a0bb1639a193bc1830f0a773e03)
All of this leads to Mackey's theorem, which is one of the central theorems in the theory of dual systems. In short, it states the bounded subsets are the same for any two Hausdorff locally convex topologies that are compatible with the same duality.
Mackey's theorem — Suppose that
is a Hausdorff locally convex space with continuous dual space
and consider the canonical duality
If
is any topology on
that is compatible with the duality
on
then the bounded subsets of
are the same as the bounded subsets of
Space of finite sequences
Let
denote the space of all sequences of scalars
such that
for all sufficiently large
Let
and define a bilinear map
by
![{\displaystyle b\left(r_{\bullet },s_{\bullet }\right):=\sum _{i=1}^{\infty }r_{i}s_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd3d240cd8c6c78e182fefe55483c2d9ec5572f)
Then
![{\displaystyle \sigma (X,X,b)=\tau (X,X,b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c0df2a4f3e3f7d0582e0d1f83e0d5b9f7022106)
Moreover, a subset
![{\displaystyle T\subseteq X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4e963e3d202098eac16189719b043a2a3a0517)
is
![{\displaystyle \sigma (X,X,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1591fe9d01b06abfc0555a2fae0e8ccfed43fb30)
-bounded (resp.
![{\displaystyle \beta (X,X,b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e26cb815ca9eb988e676bfff9134d9ea5d214eb)
-bounded) if and only if there exists a sequence
![{\displaystyle m_{\bullet }=\left(m_{i}\right)_{i=1}^{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/693616d3457eb51f4ffcc6f158d7a8254b921181)
of positive real numbers such that
![{\displaystyle \left|t_{i}\right|\leq m_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/215f786723ce2823c50c8f0938d45df62d4fe98a)
for all
![{\displaystyle t_{\bullet }=\left(t_{i}\right)_{i=1}^{\infty }\in T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/332e8f4f631f6c95569cd4da9ce03185b7c263c8)
and all indices
![{\displaystyle i}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20)
(resp. and
![{\displaystyle m_{\bullet }\in X}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29e2b313d4182961c0dd66d20206bab2152b2f2d)
).
It follows that there are weakly bounded (that is,
-bounded) subsets of
that are not strongly bounded (that is, not
-bounded).
See also
Notes
- ^ A subset
of
is total if for all
, ![{\displaystyle b(s,y)=0\quad {\text{ for all }}s\in S}](https://wikimedia.org/api/rest_v1/media/math/render/svg/256e0fde813cfe6fe2c67fc2f5dec05dadec1360)
implies
. - ^ That
is linear in its first coordinate is obvious. Suppose
is a scalar. Then
which shows that
is linear in its second coordinate. - ^ The weak topology on
is the weakest TVS topology on
making all maps
continuous, as
ranges over
The dual notation of
or simply
may also be used to denote
endowed with the weak topology
If
is not clear from context then it should be assumed to be all of
in which case it is simply called the weak topology on
(induced by
). - ^ If
is a linear map then
's transpose,
is well-defined if and only if
distinguishes points of
and
In this case, for each
the defining condition for
is:
- ^ If
is a linear map then
's transpose,
is well-defined if and only if
distinguishes points of
and
In this case, for each
the defining condition for
is:
- ^ If
is a linear map then
's transpose,
is well-defined if and only if
distinguishes points of
and
In this case, for each
the defining condition for
is:
- ^ If
is a linear map then
's transpose,
is well-defined if and only if
distinguishes points of
and
In this case, for each
the defining condition for
is:
- ^ Of course, there is an analogous definition for topologies on
to be "compatible it a pairing" but this article will only deal with topologies on
- ^ Recall that a collection of subsets of a set
is said to cover
if every point of
is contained in some set belonging to the collection.
References
Bibliography
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- Michael Reed and Barry Simon, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6.
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
- Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
- Schmitt, Lothar M (1992). "An Equivariant Version of the Hahn–Banach Theorem". Houston J. Of Math. 18: 429–447.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
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