Locally convex topological vector space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space for which the canonical evaluation map from
into its bidual (which is the strong dual of the strong dual of
) is a homeomorphism (or equivalently, a TVS isomorphism).
A normed space is reflexive if and only if this canonical evaluation map is surjective, in which case this (always linear) evaluation map is an isometric isomorphism and the normed space is a Banach space. Those spaces for which the canonical evaluation map is surjective are called semi-reflexive spaces.
In 1951, R. C. James discovered a Banach space, now known as James' space, that is not reflexive (meaning that the canonical evaluation map is not an isomorphism) but is nevertheless isometrically isomorphic to its bidual (any such isometric isomorphism is necessarily not the canonical evaluation map). So importantly, for a Banach space to be reflexive, it is not enough for it to be isometrically isomorphic to its bidual; it is the canonical evaluation map in particular that has to be a homeomorphism.
Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. Reflexive Banach spaces are often characterized by their geometric properties.
Definition
- Definition of the bidual
Suppose that
is a topological vector space (TVS) over the field
(which is either the real or complex numbers) whose continuous dual space,
separates points on
(that is, for any
there exists some
such that
).
Let
(some texts write
) denote the strong dual of
which is the vector space
of continuous linear functionals on
endowed with the topology of uniform convergence on bounded subsets of
;
this topology is also called the strong dual topology and it is the "default" topology placed on a continuous dual space (unless another topology is specified).
If
is a normed space, then the strong dual of
is the continuous dual space
with its usual norm topology.
The bidual of
denoted by
is the strong dual of
; that is, it is the space
If
is a normed space, then
is the continuous dual space of the Banach space
with its usual norm topology.
- Definitions of the evaluation map and reflexive spaces
For any
let
be defined by
where
is a linear map called the evaluation map at
;
since
is necessarily continuous, it follows that
Since
separates points on
the linear map
defined by
is injective where this map is called the evaluation map or the canonical map.
Call
semi-reflexive if
is bijective (or equivalently, surjective) and we call
reflexive if in addition
is an isomorphism of TVSs.A normable space is reflexive if and only if it is semi-reflexive or equivalently, if and only if the evaluation map is surjective.
Reflexive Banach spaces
Suppose
is a normed vector space over the number field
or
(the real numbers or the complex numbers), with a norm
Consider its dual normed space
that consists of all continuous linear functionals
and is equipped with the dual norm
defined by![{\displaystyle \|f\|^{\prime }=\sup\{|f(x)|\,:\,x\in X,\ \|x\|=1\}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The dual
is a normed space (a Banach space to be precise), and its dual normed space
is called bidual space for
The bidual consists of all continuous linear functionals
and is equipped with the norm
dual to
Each vector
generates a scalar function
by the formula:
and
is a continuous linear functional on
that is,
One obtains in this way a map
called evaluation map, that is linear. It follows from the Hahn–Banach theorem that
is injective and preserves norms:
that is,
maps
isometrically onto its image
in
Furthermore, the image
is closed in
but it need not be equal to ![{\displaystyle X^{\prime \prime }.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A normed space
is called reflexive if it satisfies the following equivalent conditions:
- the evaluation map
is surjective, - the evaluation map
is an isometric isomorphism of normed spaces, - the evaluation map
is an isomorphism of normed spaces.
A reflexive space
is a Banach space, since
is then isometric to the Banach space ![{\displaystyle X^{\prime \prime }.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Remark
A Banach space
is reflexive if it is linearly isometric to its bidual under this canonical embedding
James' space is an example of a non-reflexive space which is linearly isometric to its bidual. Furthermore, the image of James' space under the canonical embedding
has codimension one in its bidual.[2]A Banach space
is called quasi-reflexive (of order
) if the quotient
has finite dimension ![{\displaystyle d.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Examples
- Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection
from the definition is bijective, by the rank–nullity theorem. - The Banach space
of scalar sequences tending to 0 at infinity, equipped with the supremum norm, is not reflexive. It follows from the general properties below that
and
are not reflexive, because
is isomorphic to the dual of
and
is isomorphic to the dual of ![{\displaystyle \ell ^{1}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- All Hilbert spaces are reflexive, as are the Lp spaces
for
More generally: all uniformly convex Banach spaces are reflexive according to the Milman–Pettis theorem. The
and
spaces are not reflexive (unless they are finite dimensional, which happens for example when
is a measure on a finite set). Likewise, the Banach space
of continuous functions on
is not reflexive. - The spaces
of operators in the Schatten class on a Hilbert space
are uniformly convex, hence reflexive, when
When the dimension of
is infinite, then
(the trace class) is not reflexive, because it contains a subspace isomorphic to
and
(the bounded linear operators on
) is not reflexive, because it contains a subspace isomorphic to
In both cases, the subspace can be chosen to be the operators diagonal with respect to a given orthonormal basis of ![{\displaystyle H.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Properties
Since every finite-dimensional normed space is a reflexive Banach space, only infinite-dimensional spaces can be non-reflexive.
If a Banach space
is isomorphic to a reflexive Banach space
then
is reflexive.[3]
Every closed linear subspace of a reflexive space is reflexive. The continuous dual of a reflexive space is reflexive. Every quotient of a reflexive space by a closed subspace is reflexive.[4]
Let
be a Banach space. The following are equivalent.
- The space
is reflexive. - The continuous dual of
is reflexive.[5] - The closed unit ball of
is compact in the weak topology. (This is known as Kakutani's Theorem.) - Every bounded sequence in
has a weakly convergent subsequence.[7] - The statement of Riesz's lemma holds when the real number[note 1] is exactly
Explicitly, for every closed proper vector subspace
of
there exists some vector
of unit norm
such that
for all
- Using
to denote the distance between the vector
and the set
this can be restated in simpler language as:
is reflexive if and only if for every closed proper vector subspace
there is some vector
on the unit sphere of
that is always at least a distance of
away from the subspace. - For example, if the reflexive Banach space
is endowed with the usual Euclidean norm and
is the
plane then the points
satisfy the conclusion
If
is instead the
-axis then every point belonging to the unit circle in the
plane satisfies the conclusion.
- Every continuous linear functional on
attains its supremum on the closed unit ball in
[9] (James' theorem)
Since norm-closed convex subsets in a Banach space are weakly closed,[10]
it follows from the third property that closed bounded convex subsets of a reflexive space
are weakly compact. Thus, for every decreasing sequence of non-empty closed bounded convex subsets of
the intersection is non-empty. As a consequence, every continuous convex function
on a closed convex subset
of
such that the set
is non-empty and bounded for some real number
attains its minimum value on ![{\displaystyle C.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
The promised geometric property of reflexive Banach spaces is the following: if
is a closed non-empty convex subset of the reflexive space
then for every
there exists a
such that
minimizes the distance between
and points of
This follows from the preceding result for convex functions, applied to
Note that while the minimal distance between
and
is uniquely defined by
the point
is not. The closest point
is unique when
is uniformly convex.
A reflexive Banach space is separable if and only if its continuous dual is separable. This follows from the fact that for every normed space
separability of the continuous dual
implies separability of
[11]
Super-reflexive space
Informally, a super-reflexive Banach space
has the following property: given an arbitrary Banach space
if all finite-dimensional subspaces of
have a very similar copy sitting somewhere in
then
must be reflexive. By this definition, the space
itself must be reflexive. As an elementary example, every Banach space
whose two dimensional subspaces are isometric to subspaces of
satisfies the parallelogram law, hence[12]
is a Hilbert space, therefore
is reflexive. So
is super-reflexive.
The formal definition does not use isometries, but almost isometries. A Banach space
is finitely representable[13]
in a Banach space
if for every finite-dimensional subspace
of
and every
there is a subspace
of
such that the multiplicative Banach–Mazur distance between
and
satisfies![{\displaystyle d\left(X_{0},Y_{0}\right)<1+\varepsilon .}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A Banach space finitely representable in
is a Hilbert space. Every Banach space is finitely representable in
The Lp space
is finitely representable in ![{\displaystyle \ell ^{p}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
A Banach space
is super-reflexive if all Banach spaces
finitely representable in
are reflexive, or, in other words, if no non-reflexive space
is finitely representable in
The notion of ultraproduct of a family of Banach spaces[14]
allows for a concise definition: the Banach space
is super-reflexive when its ultrapowers are reflexive.
James proved that a space is super-reflexive if and only if its dual is super-reflexive.[13]
Finite trees in Banach spaces
One of James' characterizations of super-reflexivity uses the growth of separated trees.[15]The description of a vectorial binary tree begins with a rooted binary tree labeled by vectors: a tree of height
in a Banach space
is a family of
vectors of
that can be organized in successive levels, starting with level 0 that consists of a single vector
the root of the tree, followed, for
by a family of
2 vectors forming level ![{\displaystyle k:}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
that are the children of vertices of level
In addition to the tree structure, it is required here that each vector that is an internal vertex of the tree be the midpoint between its two children:![{\displaystyle x_{\emptyset }={\frac {x_{1}+x_{-1}}{2}},\quad x_{\varepsilon _{1},\ldots ,\varepsilon _{k}}={\frac {x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}+x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}}{2}},\quad 1\leq k<n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Given a positive real number
the tree is said to be
-separated if for every internal vertex, the two children are
-separated in the given space norm:![{\displaystyle \left\|x_{1}-x_{-1}\right\|\geq t,\quad \left\|x_{\varepsilon _{1},\ldots ,\varepsilon _{k},1}-x_{\varepsilon _{1},\ldots ,\varepsilon _{k},-1}\right\|\geq t,\quad 1\leq k<n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Theorem.[15]The Banach space
is super-reflexive if and only if for every
there is a number
such that every
-separated tree contained in the unit ball of
has height less than ![{\displaystyle n(t).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Uniformly convex spaces are super-reflexive.[15]Let
be uniformly convex, with modulus of convexity
and let
be a real number in
By the properties of the modulus of convexity, a
-separated tree of height
contained in the unit ball, must have all points of level
contained in the ball of radius
By induction, it follows that all points of level
are contained in the ball of radius![{\displaystyle \left(1-\delta _{X}(t)\right)^{j},\ j=1,\ldots ,n.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
If the height
was so large that
then the two points
of the first level could not be
-separated, contrary to the assumption. This gives the required bound
function of
only.
Using the tree-characterization, Enflo proved[16]
that super-reflexive Banach spaces admit an equivalent uniformly convex norm. Trees in a Banach space are a special instance of vector-valued martingales. Adding techniques from scalar martingale theory, Pisier improved Enflo's result by showing[17] that a super-reflexive space
admits an equivalent uniformly convex norm for which the modulus of convexity satisfies, for some constant
and some real number ![{\displaystyle q\geq 2,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \delta _{X}(t)\geq c\,t^{q},\quad {\text{ whenever }}t\in [0,2].}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Reflexive locally convex spaces
The notion of reflexive Banach space can be generalized to topological vector spaces in the following way.
Let
be a topological vector space over a number field
(of real numbers
or complex numbers
). Consider its strong dual space
which consists of all continuous linear functionals
and is equipped with the strong topology
that is,, the topology of uniform convergence on bounded subsets in
The space
is a topological vector space (to be more precise, a locally convex space), so one can consider its strong dual space
which is called the strong bidual space for
It consists of all continuous linear functionals
and is equipped with the strong topology
Each vector
generates a map
by the following formula:
This is a continuous linear functional on
that is,,
This induces a map called the evaluation map:
This map is linear. If
is locally convex, from the Hahn–Banach theorem it follows that
is injective and open (that is, for each neighbourhood of zero
in
there is a neighbourhood of zero
in
such that
). But it can be non-surjective and/or discontinuous.
A locally convex space
is called
- semi-reflexive if the evaluation map
is surjective (hence bijective), - reflexive if the evaluation map
is surjective and continuous (in this case
is an isomorphism of topological vector spaces[18]).
Theorem — A locally convex space
is reflexive if and only if it is semi-reflexive and barreled.
Theorem — The strong dual of a semireflexive space is barrelled.
Semireflexive spaces
Characterizations
If
is a Hausdorff locally convex space then the following are equivalent:
is semireflexive;- The weak topology on
had the Heine-Borel property (that is, for the weak topology
every closed and bounded subset of
is weakly compact). - If linear form on
that continuous when
has the strong dual topology, then it is continuous when
has the weak topology;
is barreled;
with the weak topology
is quasi-complete.
Characterizations of reflexive spaces
If
is a Hausdorff locally convex space then the following are equivalent:
is reflexive;
is semireflexive and infrabarreled;
is semireflexive and barreled;
is barreled and the weak topology on
had the Heine-Borel property (that is, for the weak topology
every closed and bounded subset of
is weakly compact).
is semireflexive and quasibarrelled.
If
is a normed space then the following are equivalent:
is reflexive;- The closed unit ball is compact when
has the weak topology ![{\displaystyle \sigma \left(X,X^{\prime }\right).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is a Banach space and
is reflexive.- Every sequence
with
for all
of nonempty closed bounded convex subsets of
has nonempty intersection.
Theorem — A real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane.
Sufficient conditions
- Normed spaces
A normed space that is semireflexive is a reflexive Banach space.A closed vector subspace of a reflexive Banach space is reflexive.
Let
be a Banach space and
a closed vector subspace of
If two of
and
are reflexive then they all are. This is why reflexivity is referred to as a three-space property.
- Topological vector spaces
If a barreled locally convex Hausdorff space is semireflexive then it is reflexive.
The strong dual of a reflexive space is reflexive.Every Montel space is reflexive. And the strong dual of a Montel space is a Montel space (and thus is reflexive).
Properties
A locally convex Hausdorff reflexive space is barrelled.
If
is a normed space then
is an isometry onto a closed subspace of
This isometry can be expressed by:![{\displaystyle \|x\|=\sup _{\stackrel {x^{\prime }\in X^{\prime },}{\|x^{\prime }\|\leq 1}}\left|\left\langle x^{\prime },x\right\rangle \right|.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Suppose that
is a normed space and
is its bidual equipped with the bidual norm. Then the unit ball of
is dense in the unit ball
of
for the weak topology ![{\displaystyle \sigma \left(X^{\prime \prime },X^{\prime }\right).}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Examples
- Every finite-dimensional Hausdorff topological vector space is reflexive, because
is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space. - A normed space
is reflexive as a normed space if and only if it is reflexive as a locally convex space. This follows from the fact that for a normed space
its dual normed space
coincides as a topological vector space with the strong dual space
As a corollary, the evaluation map
coincides with the evaluation map
and the following conditions become equivalent:
is a reflexive normed space (that is,
is an isomorphism of normed spaces),
is a reflexive locally convex space (that is,
is an isomorphism of topological vector spaces[18]),
is a semi-reflexive locally convex space (that is,
is surjective).
- A (somewhat artificial) example of a semi-reflexive space that is not reflexive is obtained as follows: let
be an infinite dimensional reflexive Banach space, and let
be the topological vector space
that is, the vector space
equipped with the weak topology. Then the continuous dual of
and
are the same set of functionals, and bounded subsets of
(that is, weakly bounded subsets of
) are norm-bounded, hence the Banach space
is the strong dual of
Since
is reflexive, the continuous dual of
is equal to the image
of
under the canonical embedding
but the topology on
(the weak topology of
) is not the strong topology
that is equal to the norm topology of ![{\displaystyle Y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- Montel spaces are reflexive locally convex topological vector spaces. In particular, the following functional spaces frequently used in functional analysis are reflexive locally convex spaces:[32]
- the space
of smooth functions on arbitrary (real) smooth manifold
and its strong dual space
of distributions with compact support on ![{\displaystyle M,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- the space
of smooth functions with compact support on arbitrary (real) smooth manifold
and its strong dual space
of distributions on ![{\displaystyle M,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- the space
of holomorphic functions on arbitrary complex manifold
and its strong dual space
of analytic functionals on ![{\displaystyle M,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- the Schwartz space
on
and its strong dual space
of tempered distributions on ![{\displaystyle \mathbb {R} ^{n}.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Counter-examples
- There exists a non-reflexive locally convex TVS whose strong dual is reflexive.
Other types of reflexivity
A stereotype space, or polar reflexive space, is defined as a topological vector space (TVS) satisfying a similar condition of reflexivity, but with the topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space
More precisely, a TVS
is called polar reflexive[34] or stereotype if the evaluation map into the second dual space
is an isomorphism of topological vector spaces.[18] Here the stereotype dual space
is defined as the space of continuous linear functionals
endowed with the topology of uniform convergence on totally bounded sets in
(and the stereotype second dual space
is the space dual to
in the same sense).
In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), it forms a closed monoidal category, and it admits standard operations (defined inside of Ste) of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category Ste have applications in duality theory for non-commutative groups.
Similarly, one can replace the class of bounded (and totally bounded) subsets in
in the definition of dual space
by other classes of subsets, for example, by the class of compact subsets in
– the spaces defined by the corresponding reflexivity condition are called reflective,[35][36] and they form an even wider class than Ste, but it is not clear (2012), whether this class forms a category with properties similar to those of Ste.
See also
- Grothendieck space
- A generalization which has some of the properties of reflexive spaces and includes many spaces of practical importance is the concept of Grothendieck space.
- Reflexive operator algebra – operator algebra that has enough invariant subspaces to characterize itPages displaying wikidata descriptions as a fallback
References
Notes
- ^ The statement of Riesz's lemma involves only one real number, which is denoted by
in the article on Riesz's lemma. The lemma always holds for all real
But for a Banach space, the lemma holds for all
if and only if the space is reflexive.
Citations
- ^ Robert C. James (1951). "A non-reflexive Banach space isometric with its second conjugate space". Proc. Natl. Acad. Sci. U.S.A. 37 (3): 174–177. Bibcode:1951PNAS...37..174J. doi:10.1073/pnas.37.3.174. PMC 1063327. PMID 16588998.
- ^ Proposition 1.11.8 in Megginson (1998, p. 99).
- ^ Megginson (1998, pp. 104–105).
- ^ Corollary 1.11.17, p. 104 in Megginson (1998).
- ^ Since weak compactness and weak sequential compactness coincide by the Eberlein–Šmulian theorem.
- ^ Theorem 1.13.11 in Megginson (1998, p. 125).
- ^ Theorem 2.5.16 in Megginson (1998, p. 216).
- ^ Theorem 1.12.11 and Corollary 1.12.12 in Megginson (1998, pp. 112–113).
- ^ see this characterization of Hilbert space among Banach spaces
- ^ a b James, Robert C. (1972), "Super-reflexive Banach spaces", Can. J. Math. 24:896–904.
- ^ Dacunha-Castelle, Didier; Krivine, Jean-Louis (1972), "Applications des ultraproduits à l'étude des espaces et des algèbres de Banach" (in French), Studia Math. 41:315–334.
- ^ a b c see James (1972).
- ^ Enflo, Per (1972). "Banach spaces which can be given an equivalent uniformly convex norm". Israel Journal of Mathematics. 13: 281–288. doi:10.1007/BF02762802.
- ^ Pisier, Gilles (1975). "Martingales with values in uniformly convex spaces". Israel Journal of Mathematics. 20: 326–350. doi:10.1007/BF02760337.
- ^ a b c An isomorphism of topological vector spaces is a linear and a homeomorphic map
![{\displaystyle \varphi :X\to Y.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- ^ Edwards 1965, 8.4.7.
- ^ Köthe, Gottfried (1983). Topological Vector Spaces I. Springer Grundlehren der mathematischen Wissenschaften. Springer. ISBN 978-3-642-64988-2.
- ^ Garibay Bonales, F.; Trigos-Arrieta, F. J.; Vera Mendoza, R. (2002). "A characterization of Pontryagin-van Kampen duality for locally convex spaces". Topology and Its Applications. 121 (1–2): 75–89. doi:10.1016/s0166-8641(01)00111-0.
- ^ Akbarov, S. S.; Shavgulidze, E. T. (2003). "On two classes of spaces reflexive in the sense of Pontryagin". Mat. Sbornik. 194 (10): 3–26.
General references
- Bernardes, Nilson C. Jr. (2012), On nested sequences of convex sets in Banach spaces, vol. 389, Journal of Mathematical Analysis and Applications, pp. 558–561 .
- Conway, John B. (1985). A Course in Functional Analysis. Springer.
- Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
- Edwards, R. E. (1965). Functional analysis. Theory and applications. New York: Holt, Rinehart and Winston. ISBN 0030505356.
- James, Robert C. (1972), Some self-dual properties of normed linear spaces. Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967), Ann. of Math. Studies, vol. 69, Princeton, NJ: Princeton Univ. Press, pp. 159–175.
- Khaleelulla, S. M. (1982). Counterexamples in Topological Vector Spaces. Lecture Notes in Mathematics. Vol. 936. Berlin, Heidelberg, New York: Springer-Verlag. ISBN 978-3-540-11565-6. OCLC 8588370.
- Kolmogorov, A. N.; Fomin, S. V. (1957). Elements of the Theory of Functions and Functional Analysis, Volume 1: Metric and Normed Spaces. Rochester: Graylock Press.
- Megginson, Robert E. (1998), An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, New York: Springer-Verlag, pp. xx+596, ISBN 0-387-98431-3
- Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834.
- 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. (1966). Topological vector spaces. New York: The Macmillan Company.
- 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.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.