Infunctional andconvex analysis, and related disciplines ofmathematics, thepolar set${\displaystyle A^{\circ }}$ is a special convex set associated to any subset${\displaystyle A}$ of avector space${\displaystyle X,}$ lying in thedual space${\displaystyle X^{\mathrm{\prime }}.}$ Thebipolar of a subset is the polar of${\displaystyle A^{\circ },}$ but lies in${\displaystyle X}$ (not${\displaystyle X^{\mathrm{\prime }\mathrm{\prime }}}$).

There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.^{[1][citation needed]} In each case, the definition describes a duality between certain subsets of apairing of vector spaces${\displaystyle ⟨X,Y⟩}$ over the real or complex numbers (${\displaystyle X}$ and${\displaystyle Y}$ are oftentopological vector spaces (TVSs)).

If${\displaystyle X}$ is a vector space over the field${\displaystyle \mathbb{K}}$ then unless indicated otherwise,${\displaystyle Y}$ will usually, but not always, be some vector space oflinear functionals on${\displaystyle X}$ and the dual pairing${\displaystyle ⟨\cdot ,\cdot ⟩:X\times Y\to \mathbb{K}}$ will be thebilinearevaluation (at a point)map defined by$${\displaystyle ⟨x,f⟩:=f(x).}$$If${\displaystyle X}$ is atopological vector space then the space${\displaystyle Y}$ will usually, but not always, be thecontinuous dual space of${\displaystyle X,}$ in which case the dual pairing will again be the evaluation map.

Denote the closed ball of radius${\displaystyle r\ge 0}$ centered at the origin in the underlying scalar field${\displaystyle \mathbb{K}}$ of${\displaystyle X}$ by$${\displaystyle B_r:=B_r^{\mathbb{K}}:=\{s\in \mathbb{K}:|s|\le r\}.}$$

Suppose that${\displaystyle ⟨X,Y⟩}$ is apairing. Thepolar orabsolute polar of a subset${\displaystyle A}$ of${\displaystyle X}$ is the set:

where${\displaystyle ⟨A,y⟩:=\{⟨a,y⟩:a\in A\}}$ denotes theimage of the set${\displaystyle A}$ under the map${\displaystyle ⟨\cdot ,y⟩:X\to \mathbb{K}}$ defined by${\displaystyle x\mapsto ⟨x,y⟩.}$ If${\displaystyle \mathrm{cobal}A}$ denotes theconvex balanced hull of${\displaystyle A,}$ which by definition is the smallestconvex andbalanced subset of${\displaystyle X}$ that contains${\displaystyle A,}$ then${\displaystyle A^{\circ }=[\mathrm{cobal}A{]}^{\circ }.}$

This is anaffine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball (in${\displaystyle X}$) is precisely the unit ball (in${\displaystyle Y}$).

Theprepolar orabsolute prepolar of a subset${\displaystyle B}$ of${\displaystyle Y}$ is the set:$${\displaystyle {}^{\circ }B:=\left\{x\in X:\underset{b\in B}{sup}|⟨x,b⟩|\le 1\right\}=\{x\in X:sup|⟨x,B⟩|\le 1\}}$$

Very often, the prepolar of a subset${\displaystyle B}$ of${\displaystyle Y}$ is also called thepolar orabsolute polar of${\displaystyle B}$ and denoted by${\displaystyle B^{\circ }}$; in practice, this reuse of notation and of the word "polar" rarely causes any issues (such as ambiguity) and many authors do not even use the word "prepolar".

Thebipolar of a subset${\displaystyle A}$ of${\displaystyle X,}$ often denoted by${\displaystyle A^{\circ \circ },}$ is the set${\displaystyle {}^{\circ }\left(A^{\circ }\right)}$; that is,$${\displaystyle A^{\circ \circ }:={}^{\circ }\left(A^{\circ }\right)=\left\{x\in X:\underset{y\in A^{\circ }}{sup}|⟨x,y⟩|\le 1\right\}.}$$

Thereal polar of a subset${\displaystyle A}$ of${\displaystyle X}$ is the set:$${\displaystyle A^r:=\left\{y\in Y:\underset{a\in A}{sup}\mathrm{Re}⟨a,y⟩\le 1\right\}}$$and thereal prepolar of a subset${\displaystyle B}$ of${\displaystyle Y}$ is the set:$${\displaystyle {}^{r}B:=\left\{x\in X:\underset{b\in B}{sup}\mathrm{Re}⟨x,b⟩\le 1\right\}.}$$

As with the absolute prepolar, the real prepolar is usually called thereal polar and is also denoted by${\displaystyle B^r.}$^[2] It's important to note that some authors (e.g. [Schaefer 1999]) define "polar" to mean "real polar" (rather than "absolute polar", as is done in this article) and use the notation${\displaystyle A^{\circ }}$ for it (rather than the notation${\displaystyle A^r}$ that is used in this article and in [Narici 2011]).

Thereal bipolar of a subset${\displaystyle A}$ of${\displaystyle X,}$ sometimes denoted by${\displaystyle A^{rr},}$ is the set${\displaystyle {}^r\left(A^r\right)}$; it is equal to the${\displaystyle \sigma (X,Y)}$-closure of theconvex hull of${\displaystyle A\cup \{0\}.}$^[2]

For a subset${\displaystyle A}$ of${\displaystyle X,}$${\displaystyle A^r}$ is convex,${\displaystyle \sigma (Y,X)}$-closed, and contains${\displaystyle A^{\circ }.}$^[2] In general, it is possible that${\displaystyle A^{\circ }\ne A^r}$ but equality will hold if${\displaystyle A}$ isbalanced. Furthermore,${\displaystyle A^{\circ }=\left(\mathrm{bal}\left(A^r\right)\right)}$ where${\displaystyle \mathrm{bal}\left(A^r\right)}$ denotes thebalanced hull of${\displaystyle A^r.}$^[2]

The definition of the "polar" of a set is not universally agreed upon. Although this article defined "polar" to mean "absolute polar", some authors define "polar" to mean "real polar" and other authors use still other definitions. No matter how an author defines "polar", the notation${\displaystyle A^{\circ }}$ almost always representstheir choice of the definition (so the meaning of the notation${\displaystyle A^{\circ }}$ may vary from source to source). In particular, the polar of${\displaystyle A}$ is sometimes defined as:$${\displaystyle A^{|r|}:=\left\{y\in Y:\underset{a\in A}{sup}|\mathrm{Re}⟨a,y⟩|\le 1\right\}}$$where the notation${\displaystyle A^{|r|}}$ isnot standard notation.

We now briefly discuss how these various definitions relate to one another and when they are equivalent.

It is always the case that$${\displaystyle A^{\circ }\subseteq A^{|r|}\subseteq A^r}$$and if${\displaystyle ⟨\cdot ,\cdot ⟩}$ is real-valued (or equivalently, if${\displaystyle X}$ and${\displaystyle Y}$ are vector spaces over${\displaystyle \mathbb{R}}$) then${\displaystyle A^{\circ }=A^{|r|}.}$

If${\displaystyle A}$ is asymmetric set (that is,${\displaystyle -A=A}$ or equivalently,${\displaystyle -A\subseteq A}$) then${\displaystyle A^{|r|}=A^r}$ where if in addition${\displaystyle ⟨\cdot ,\cdot ⟩}$ is real-valued then${\displaystyle A^{\circ }=A^{|r|}=A^r.}$

If${\displaystyle X}$ and${\displaystyle Y}$ are vector spaces over${\displaystyle \mathbb{C}}$ (so that${\displaystyle ⟨\cdot ,\cdot ⟩}$ is complex-valued) and if${\displaystyle iA\subseteq A}$ (where note that this implies${\displaystyle -A=A}$ and${\displaystyle iA=A}$), then$${\displaystyle A^{\circ }\subseteq A^{|r|}=A^r\subseteq {\left(\frac{1}{\sqrt{2}}A\right)}^{\circ }}$$where if in addition${\displaystyle e^{ir}A\subseteq A}$ for all real${\displaystyle r}$ then${\displaystyle A^{\circ }=A^r.}$

Thus for all of these definitions of the polar set of${\displaystyle A}$ to agree, it suffices that${\displaystyle sA\subseteq A}$ for all scalars${\displaystyle s}$ ofunit length^{[note 1]} (where this is equivalent to${\displaystyle sA=A}$ for all unit length scalar${\displaystyle s}$).In particular, all definitions of the polar of${\displaystyle A}$ agree when${\displaystyle A}$ is abalanced set (which is often, but not always, the case) so that often, which of these competing definitions is used is immaterial. However, these differences in the definitions of the "polar" of a set${\displaystyle A}$ do sometimes introduce subtle or important technical differences when${\displaystyle A}$ is not necessarily balanced.

Algebraic dual space

If${\displaystyle X}$ is any vector space then let${\displaystyle X^{\mathrm{\#}}}$ denote thealgebraic dual space of${\displaystyle X,}$ which is the set of alllinear functionals on${\displaystyle X.}$ The vector space${\displaystyle X^{\mathrm{\#}}}$ is always a closed subset of the space${\displaystyle {\mathbb{K}}^X}$ of all${\displaystyle \mathbb{K}}$-valued functions on${\displaystyle X}$ under the topology of pointwise convergence so when${\displaystyle X^{\mathrm{\#}}}$ is endowed with the subspace topology, then${\displaystyle X^{\mathrm{\#}}}$ becomes aHausdorffcompletelocally convextopological vector space (TVS). For any subset${\displaystyle A\subseteq X,}$ let

If${\displaystyle A\subseteq B\subseteq X}$ are any subsets then${\displaystyle B^{\mathrm{\#}}\subseteq A^{\mathrm{\#}}}$ and${\displaystyle A^{\mathrm{\#}}=[\mathrm{cobal}A{]}^{\mathrm{\#}},}$ where${\displaystyle \mathrm{cobal}A}$ denotes theconvex balanced hull of${\displaystyle A.}$ For any finite-dimensional vector subspace${\displaystyle Y}$ of${\displaystyle X,}$ let${\displaystyle {\tau }_Y}$ denote theEuclidean topology on${\displaystyle Y,}$ which is the unique topology that makes${\displaystyle Y}$ into aHausdorff topological vector space (TVS). If${\displaystyle A_{\cup \mathrm{cl}\mathrm{Finite}}}$ denotes the union of allclosures${\displaystyle {\mathrm{cl}}_{\left(Y,{\tau }_Y\right)}(Y\cap A)}$ as${\displaystyle Y}$ varies over all finite dimensional vector subspaces of${\displaystyle X,}$ then${\displaystyle A^{\mathrm{\#}}={\left[A_{\cup \mathrm{cl}\mathrm{Finite}}\right]}^{\mathrm{\#}}}$ (see this footnote^{[note 2]} for an explanation). If${\displaystyle A}$ is an absorbing subset of${\displaystyle X}$ then by theBanach–Alaoglu theorem,${\displaystyle A^{\mathrm{\#}}}$ is aweak-* compact subset of${\displaystyle X^{\mathrm{\#}}.}$

If${\displaystyle A\subseteq X}$ is any non-empty subset of a vector space${\displaystyle X}$ and if${\displaystyle Y}$ is any vector space of linear functionals on${\displaystyle X}$ (that is, a vector subspace of thealgebraic dual space of${\displaystyle X}$) then the real-valued map

${\displaystyle |\cdot {|}_A:Y\to \mathbb{R}}$     defined by    ${\displaystyle {\left|x^{\mathrm{\prime }}\right|}_A:=sup\left|x^{\mathrm{\prime }}(A)\right|:=\underset{a\in A}{sup}\left|x^{\mathrm{\prime }}(a)\right|}$

is aseminorm on${\displaystyle Y.}$ If${\displaystyle A=\varnothing }$ then by definition of thesupremum,${\displaystyle sup\left|x^{\mathrm{\prime }}(A)\right|=-\mathrm{\infty }}$ so that the map${\displaystyle |\cdot {|}_{\varnothing }=-\mathrm{\infty }}$ defined above would not be real-valued and consequently, it would not be a seminorm.

Continuous dual space

Suppose that${\displaystyle X}$ is atopological vector space (TVS) withcontinuous dual space${\displaystyle X^{\mathrm{\prime }}.}$ The important special case where${\displaystyle Y:=X^{\mathrm{\prime }}}$ and the brackets represent the canonical map:$${\displaystyle ⟨x,x^{\mathrm{\prime }}⟩:=x^{\mathrm{\prime }}(x)}$$is now considered. The triple${\displaystyle ⟨X,X^{\mathrm{\prime }}⟩}$ is the called thecanonicalpairing associated with${\displaystyle X.}$

The polar of a subset${\displaystyle A\subseteq X}$ with respect to this canonical pairing is:

For any subset${\displaystyle A\subseteq X,}$${\displaystyle A^{\circ }={\left[{\mathrm{cl}}_{X}A\right]}^{\circ }}$ where${\displaystyle {\mathrm{cl}}_{X}A}$ denotes theclosure of${\displaystyle A}$ in${\displaystyle X.}$

TheBanach–Alaoglu theorem states that if${\displaystyle A\subseteq X}$ is a neighborhood of the origin in${\displaystyle X}$ then${\displaystyle A^{\circ }=A^{\mathrm{\#}}}$ and this polar set is acompact subset of the continuous dual space${\displaystyle X^{\mathrm{\prime }}}$ when${\displaystyle X^{\mathrm{\prime }}}$ is endowed with theweak-* topology (also known as the topology of pointwise convergence).

If${\displaystyle A}$ satisfies${\displaystyle sA\subseteq A}$ for all scalars${\displaystyle s}$ of unit length then one may replace the absolute value signs by${\displaystyle \mathrm{Re}}$ (the real part operator) so that:

The prepolar of a subset${\displaystyle B}$ of${\displaystyle Y=X^{\mathrm{\prime }}}$ is:$${\displaystyle {}^{\circ }B:=\left\{x\in X:\underset{b^{\mathrm{\prime }}\in B}{sup}\left|b^{\mathrm{\prime }}(x)\right|\le 1\right\}=\{x\in X:sup|B(x)|\le 1\}}$$

If${\displaystyle B}$ satisfies${\displaystyle sB\subseteq B}$ for all scalars${\displaystyle s}$ of unit length then one may replace the absolute value signs with${\displaystyle \mathrm{Re}}$ so that:$${\displaystyle {}^{\circ }B=\left\{x\in X:\underset{b^{\mathrm{\prime }}\in B}{sup}\mathrm{Re}{b}^{\mathrm{\prime }}(x)\le 1\right\}=\{x\in X:sup\mathrm{Re}B(x)\le 1\}}$$where${\displaystyle B(x):=\left\{b^{\mathrm{\prime }}(x):b^{\mathrm{\prime }}\in B\right\}.}$

Thebipolar theorem characterizes the bipolar of a subset of a topological vector space.

If${\displaystyle X}$ is a normed space and${\displaystyle S}$ is the open or closed unit ball in${\displaystyle X}$ (or even any subset of the closed unit ball that contains the open unit ball) then${\displaystyle S^{\circ }}$ is the closed unit ball in the continuous dual space${\displaystyle X^{\mathrm{\prime }}}$ when${\displaystyle X^{\mathrm{\prime }}}$ is endowed with its canonicaldual norm.

Thepolar cone of a convex cone${\displaystyle A\subseteq X}$ is the set$${\displaystyle A^{\circ }:=\left\{y\in Y:\underset{x\in A}{sup}⟨x,y⟩\le 0\right\}}$$

This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point${\displaystyle x\in X}$ is the locus${\displaystyle \{y:⟨y,x⟩=0\}}$; thedual relationship for a hyperplane yields that hyperplane's polar point.^{[3][citation needed]}

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.^[4]

Unless stated otherwise,${\displaystyle ⟨X,Y⟩}$ will be apairing. The topology${\displaystyle \sigma (Y,X)}$ is theweak-* topology on${\displaystyle Y}$ while${\displaystyle \sigma (X,Y)}$ is theweak topology on${\displaystyle X.}$ For any set${\displaystyle A,}$${\displaystyle A^r}$ denotes the real polar of${\displaystyle A}$ and${\displaystyle A^{\circ }}$ denotes the absolute polar of${\displaystyle A.}$ The term "polar" will refer to theabsolute polar.

• The (absolute) polar of a set isconvex andbalanced.^[5]
• The real polar${\displaystyle A^r}$ of a subset${\displaystyle A}$ of${\displaystyle X}$ is convex butnot necessarily balanced;${\displaystyle A^r}$ will be balanced if${\displaystyle A}$ is balanced.^[6]
• If${\displaystyle sA\subseteq A}$ for all scalars${\displaystyle s}$ of unit length then${\displaystyle A^{\circ }=A^r.}$
• ${\displaystyle A^{\circ }}$ isclosed in${\displaystyle Y}$ under theweak-*-topology on${\displaystyle Y}$.^[3]
• A subset${\displaystyle S}$ of${\displaystyle X}$ is weakly bounded (i.e.${\displaystyle \sigma (X,Y)}$-bounded) if and only if${\displaystyle S^{\circ }}$ isabsorbing in${\displaystyle Y}$.^[2]
• For a dual pair${\displaystyle ⟨X,X^{\mathrm{\prime }}⟩,}$ where${\displaystyle X}$ is a TVS and${\displaystyle X^{\mathrm{\prime }}}$ is its continuous dual space, if${\displaystyle B\subseteq X}$ is bounded then${\displaystyle B^{\circ }}$ isabsorbing in${\displaystyle X^{\mathrm{\prime }}.}$^[5] If${\displaystyle X}$ is locally convex and${\displaystyle B^{\circ }}$ is absorbing in${\displaystyle X^{\mathrm{\prime }}}$ then${\displaystyle B}$ is bounded in${\displaystyle X.}$ Moreover, a subset${\displaystyle S}$ of${\displaystyle X}$ is weakly bounded if and only if${\displaystyle S^{\circ }}$ isabsorbing in${\displaystyle X^{\mathrm{\prime }}.}$
• The bipolar${\displaystyle A^{\circ \circ }}$ of a set${\displaystyle A}$ is the${\displaystyle \sigma (X,Y)}$-closed convex hull of${\displaystyle A\cup \{0\},}$ that is the smallest${\displaystyle \sigma (X,Y)}$-closed and convex set containing both${\displaystyle A}$ and${\displaystyle 0.}$
  • Similarly, the bidual cone of a cone${\displaystyle A}$ is the${\displaystyle \sigma (X,Y)}$-closedconic hull of${\displaystyle A.}$^[7]
• If${\displaystyle \mathcal{B}}$ is a base at the origin for a TVS${\displaystyle X}$ then${\displaystyle X^{\mathrm{\prime }}=\bigcup _{B\in \mathbb{B}}\left(B^{\circ }\right).}$^[8]
• If${\displaystyle X}$ is a locally convex TVS then the polars (taken with respect to${\displaystyle ⟨X,X^{\mathrm{\prime }}⟩}$) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of${\displaystyle X^{\mathrm{\prime }}}$ (i.e. given any bounded subset${\displaystyle H}$ of${\displaystyle X_{\sigma }^{\mathrm{\prime }},}$ there exists a neighborhood${\displaystyle S}$ of the origin in${\displaystyle X}$ such that${\displaystyle H\subseteq S^{\circ }}$).^[6]
  • Conversely, if${\displaystyle X}$ is a locally convex TVS then the polars (taken with respect to${\displaystyle ⟨X,X^{\mathrm{\#}}⟩}$) of any fundamental family of equicontinuous subsets of${\displaystyle X^{\mathrm{\prime }}}$ form a neighborhood base of the origin in${\displaystyle X.}$^[6]
• Let${\displaystyle X}$ be a TVS with a topology${\displaystyle \tau .}$ Then${\displaystyle \tau }$ is a locally convex TVS topology if and only if${\displaystyle \tau }$ is the topology of uniform convergence on the equicontinuous subsets of${\displaystyle X^{\mathrm{\prime }}.}$^[6]

The last two results explain why equicontinuous subsets of the continuous dual space play such a prominent role in the modern theory of functional analysis: because equicontinuous subsets encapsulate all information about the locally convex space${\displaystyle X}$'s original topology.

Set relations

• ${\displaystyle X^{\circ }=X^{|r|}=X^r=\{0\}}$^[6] and${\displaystyle {\varnothing }^{\circ }={\varnothing }^{|r|}={\varnothing }^r=Y.}$
• For all scalars${\displaystyle s\ne 0,}$${\displaystyle (sA{)}^{\circ }=\frac{1}{s}\left(A^{\circ }\right)}$ and for all real${\displaystyle t\ne 0,}$${\displaystyle (tA{)}^{|r|}=\frac{1}{t}\left(A^{|r|}\right)}$ and${\displaystyle (tA{)}^r=\frac{1}{t}\left(A^r\right).}$
• ${\displaystyle A^{\circ \circ \circ }=A^{\circ }.}$ However, for the real polar we have${\displaystyle A^{rrr}\subseteq A^r.}$^[6]
• For any finite collection of sets${\displaystyle A_1,\dots ,A_n,}$$${\displaystyle {\left(A_1\cap \cdots \cap A_n\right)}^{\circ }=\left(A_1^{\circ }\right)\cup \cdots \cup \left(A_n^{\circ }\right).}$$
• If${\displaystyle A\subseteq B}$ then${\displaystyle B^{\circ }\subseteq A^{\circ },}$${\displaystyle B^r\subseteq A^r,}$ and${\displaystyle B^{|r|}\subseteq A^{|r|}.}$
  • An immediate corollary is that${\displaystyle \bigcup _{i\in I}\left(A_i^{\circ }\right)\subseteq {\left(\bigcap _{i\in I}{A}_i\right)}^{\circ }}$; equality necessarily holds when${\displaystyle I}$ is finite and may fail to hold if${\displaystyle I}$ is infinite.
• ${\displaystyle \bigcap _{i\in I}\left(A_i^{\circ }\right)={\left(\bigcup _{i\in I}{A}_i\right)}^{\circ }}$ and${\displaystyle \bigcap _{i\in I}\left(A_i^r\right)={\left(\bigcup _{i\in I}{A}_i\right)}^r.}$
• If${\displaystyle C}$ is a cone in${\displaystyle X}$ then${\displaystyle C^{\circ }=\left\{y\in Y:⟨c,y⟩=0\text{ for all }c\in C\right\}.}$^[5]
• If${\displaystyle {\left(S_i\right)}_{i\in I}}$ is a family of${\displaystyle \sigma (X,Y)}$-closed subsets of${\displaystyle X}$ containing${\displaystyle 0\in X,}$ then the real polar of${\displaystyle {\cap }_{i\in I}{S}_i}$ is the closed convex hull of${\displaystyle {\cup }_{i\in I}\left(S_i^r\right).}$^[6]
• If${\displaystyle 0\in A\cap B}$ then${\displaystyle A^{\circ }\cap B^{\circ }\subseteq 2\left[(A+B{)}^{\circ }\right]\subseteq 2\left(A^{\circ }\cap B^{\circ }\right).}$^[9]
• For a closedconvex cone${\displaystyle C}$ in a real vector space${\displaystyle X,}$ thepolar cone is the polar of${\displaystyle C}$; that is,$${\displaystyle C^{\circ }=\{y\in Y:\underset{}{sup}⟨C,y⟩\le 0\},}$$ where${\displaystyle \underset{}{sup}⟨C,y⟩:=\underset{c\in C}{sup}⟨c,y⟩.}$^[1]

• Banach–Alaoglu theorem – Theorem in functional analysis
• Bipolar theorem – Theorem in convex analysis
• Polar cone – Concepts in convex analysisPages displaying short descriptions of redirect targets
• Polar topology – Dual space topology of uniform convergence on some sub-collection of bounded subsets
• Locally convex topological vector space – Vector space with a topology defined by convex open sets
• Topological vector space – Vector space with a notion of nearness

• Jarchow, Hans (1981).Locally convex spaces. Stuttgart: B.G. Teubner.ISBN 978-3-519-02224-4.OCLC 8210342.
• Köthe, Gottfried (1983) [1969].Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media.ISBN 978-3-642-64988-2.MR 0248498.OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN 978-1584888666.OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980).Topological Vector Spaces.Cambridge Tracts in Mathematics. Vol. 53. Cambridge England:Cambridge University Press.ISBN 978-0-521-29882-7.OCLC 589250.
• 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.
• Wilansky, Albert (2013).Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc.ISBN 978-0-486-49353-4.OCLC 849801114.

• Functional analysis
• Linear functionals
• Topological vector spaces

• Articles with short description
• Short description with empty Wikidata description
• All articles with unsourced statements
• Articles with unsourced statements from September 2018
• Pages displaying short descriptions of redirect targets via Module:Annotated link