In functional analysis and related areas of mathematics a polar topology, topology of 𝒢convergence or topology of uniform convergence on the sets of 𝒢 is a method to define locally convex topologies on the vector spaces of a pairing.
Preliminaries
A pairing is a triple consisting of two vector spaces over a field 𝔽 (either the real or complex numbers) and a bilinear map b : X × Y → 𝔽. A dual pair or dual system is a pairing satisfying the following two separation axioms:
 Y separates/distinguishes points of X: for all nonzero x ∈ X, there exists y ∈ Y such that b(x, y) ≠ 0, and
 X separates/distinguishes points of Y: for all nonzero y ∈ Y, there exists x ∈ X such that b(x, y) ≠ 0.
Polars
The polar or absolute polar of a subset A ⊆ X is the set^{[1]}
Dually, the polar or absolute polar of a subset B ⊆Y is denoted by B° and defined by
In this case, the absolute polar of a subset B ⊆Y is also called the prepolar of B and may be denoted by °B.
The polar is a convex balanced set containing the origin.^{[2]}
If A ⊆ X then the bipolar of A, denoted by A°°, is defined by A°° = (A^{⊥}). Similarly, if B ⊆ Y then the bipolar of B is defined to be B°° = (°B)°.
Weak topologies
Suppose that is a pairing of vector spaces over 𝕂.
 Notation: For all x ∈ X, let b(x, •) : Y → 𝕂 denote the linear functional on Y defined by y ↦ b(x, y) and let b(X, •) = { b(x, •) : x ∈ X }.
 Similarly, for all y ∈ Y, let b(•, y) : X → 𝕂 be defined by x ↦ b(x, y) and let b(•, Y) = { b(•, y) : y ∈ Y }.
The weak topology on X induced by Y (and b) is the weakest TVS topology on X, denoted by or simply making all maps b( •, y) : X → 𝕂 continuous, as y ranges over Y.^{[3]} Similarly, there are the dual definition of the weak topology on Y induced by X (and b), which is denoted by or simply : it is the weakest TVS topology on Y making all maps b(x, •) : Y → 𝕂 continuous, as x ranges over X.^{[3]}
Weak boundedness and absorbing polars
It is because of the following theorem that it is almost always assumed that the family 𝒢 consists of bounded subsets of X.^{[3]}
Theorem — For any subset A ⊆X, the following are equivalent:
 A° is an absorbing subset of Y.
 If this condition is not satisfied then A° can not possibly be a neighborhood of the origin in any TVS topology on X';
 A is a bounded set; said differently, A is a bounded subset of (X, 𝜎(X, Y, b));
 for all y ∈ Y, where this supremum may also be denoted by
The bounded subsets of Y have an analogous characterization.
Dual definitions and results
Every pairing can be associated with a corresponding pairing where by definition ^{[3]}
There is a repeating theme in duality theory, which is 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 If the definition depends on the order of X and Y (e.g. the definition of "the weak topology defined on X by Y") then by switching the order of X and Y, it is meant that this definition shuold be applied to (e.g. this gives us the definition of "the weak topology defined on Y by X").
For instance, after defining "X distinguishes points of Y" (resp, "S is a total subset of Y") as above, then the dual definition of "Y distinguishes points of X" (resp, "S is a total subset of X") is immediately obtained. For instance, once is defined then it should be automatically assume that has been defined without mentioning the analogous definition. The same applies to many theorems.
 Convention: Adhering to common practice, unless clarity is needed, whenever a definition (or result) is given for a pairing then mention the corresponding dual definition (or result) will be omitted but it may nevertheless be used.
In particular, although this article will only define the general notion of polar topologies on Y with 𝒢 being a collection of bounded subsets of X, this article will nevertheless use the dual definition for polar topologies on X with 𝒢 being a collection of bounded subsets of Y.
 Identification of (X, Y) with (Y, X)
Although it is technically incorrect and an abuse of notation, the following convention is nearly ubiquitous:
 Convention: This article will use the common practice of treating a pairing interchangeably with and also denoting by
Polar topologies
Throughout, is a pairing of vector spaces over the field 𝕂 and 𝒢 is a nonempty collection of bounded subsets of X.
For every G ∈ 𝒢 and r > 0, r G° = r (G°) is convex and balanced and because G is a bounded, the set r G° is absorbing in Y.
The polar topology on Y determined (or generated) by 𝒢 (and b), also called the 𝒢topology on Y or the topology of uniform convergence on the sets of 𝒢, is the unique topological vector space (TVS) topology on Y for which
forms a neighbourhood subbasis at the origin.^{[3]} When Y is endowed with this 𝒢topology then it is denoted by Y_{𝒢}.
If is a sequence of positive numbers converging to 0 then the defining neighborhood subbasis at 0 may be replaced with
without changing the resulting topology.
When 𝒢 is a directed set with respect to subset inclusion (i.e. if for all G, H ∈ 𝒢, there exists some K ∈ 𝒢 such that G ∪ H ⊆ K) then the defining neighborhood subbasis at the origin actually forms a neighborhood basis at 0.^{[3]}
 Seminorms defining the polar topology
Every G ∈ 𝒢 determines a seminorm p_{G} : Y → ℝ defined by
where G° = { y ∈ Y : p_{G}(y) ≤ 1} and p_{G} is in fact the Minkowski functional of G°. Because of this, the 𝒢topology on Y is always a locally convex topology.^{[3]}
 Modifying 𝒢
If every positive scalar multiple of a set in 𝒢 is contained in some set belonging to 𝒢 then the defining neighborhood subbasis at the origin can be replaced with
without changing the resulting topology.
The following theorem gives ways in which 𝒢 can be modified without changing the resulting 𝒢topology on Y.
Theorem^{[3]} — Let is a pairing of vector spaces over 𝕂 and let 𝒢 be a nonempty collection of bounded subsets of X. The 𝒢topology on Y is not altered if 𝒢 is replaced by any of the following collections of [bounded] subsets of X:
 all subsets of all finite unions of sets in 𝒢;
 all scalar multiples of all sets in 𝒢;
 the balanced hull of every set in 𝒢;
 the convex hull of every set in 𝒢;
 the closure of every set in 𝒢;
 the closure of the convex balanced hull of every set in 𝒢.
It is because of this theorem that many authors often require that 𝒢 also satisfy the following additional conditions:
 The union of any two sets A, B ∈ 𝒢 is contained in some set C ∈ 𝒢;
 All scalar multiples of every G ∈ 𝒢 belongs to 𝒢.
Some authors^{[4]} further assume that every x ∈ X belongs to some set G ∈ 𝒢 because this assumption suffices to ensure that the 𝒢topology is Hausdorff.
 Convergence of nets and filters
If is a net in Y then in the 𝒢topology on Y if and only if for every G ∈ 𝒢, or in words, if and only if for every G ∈ 𝒢, the net of linear functionals on X converges uniformly to 0 on G; here, for each i ∈ I, the linear functional is defined by
If y ∈ Y then in the 𝒢topology on Y if and only if for all G ∈ 𝒢,
A filter ℱ on Y converges to an element y ∈ Y in the 𝒢topology on Y if ℱ converges uniformly to y on each G ∈ 𝒢.
Properties
 The results in the article Topologies on spaces of linear maps can be applied to polar topologies.
Throughout, is a pairing of vector spaces over the field 𝕂 and 𝒢 is a nonempty collection of bounded subsets of X.
 Hausdorffness
 We say that 𝒢 covers X if every point in X belong to some set in 𝒢.
 We say that 𝒢 is total in X^{[5]} if the linear span of is dense in X.
Theorem — Let be a pairing of vector spaces over the field 𝕂 and 𝒢 be a nonempty collection of bounded subsets of X. Then,
 If 𝒢 covers X then the 𝒢topology on Y is Hausdorff.^{[3]}
 If X distinguishes points of Y and if is a dense subset of X then the 𝒢topology on Y is Hausdorff.^{[2]}
 If is a dual system (rather than merely a pairing) then the 𝒢topology on Y is Hausdorff if and only if span of is dense in ^{[3]}
Proof


Proof of (2): If then we're done so assume otherwise. Since the 𝒢topology on Y is a TVS topology, it suffices to show that the set is closed in Y. Let y ∈ Y be nonzero, let f : X → 𝕂 be defined by f(x) =} b(x, y) for all x ∈ X, and let V = { s ∈ 𝕂 : s > 1}. Since X distinguishes points of Y, there exists some (nonzero) x ∈ X such that f(x) ≠ 0 where (since f is surjective) it can be assumed without loss of generality that f(x) > 1. The set is a open subset of X that is not empty (since it contains x). Since is a dense subset of X there exists some G ∈ 𝒢 and some g ∈ G such that g ∈ U. Since g ∈ U, b(g, y) > 1 so that y ∉ G°, where G° is a subbasic closed neighborhood of the origin in the 𝒢topology on Y. ■ 
Examples of polar topologies induced by a pairing
Throughout, will be a pairing of vector spaces over the field 𝕂 and 𝒢 will be a nonempty collection of bounded subsets of X.
The following table will omit mention of b. The topologies are listed in an order that roughly corresponds with coarser topologies first and the finer topologies last; note that some of these topologies may be out of order e.g. and the topology below it (i.e. the topology generated by complete and bounded disks) or if is not Hausdorff. If more than one collection of subsets appears the same row in the leftmost column then that means that the same polar topology is generated by these collections.
 Notation: If 𝛥(Y, X, b) denotes a polar topology on Y then Y endowed with this topology will be denoted by Y_{𝛥(Y, X, b)}, Y_{𝛥(Y, X)} or simply Y_{𝛥} (e.g. if then {{{1}}} so that Y_{σ(Y, X, b)}, Y_{σ(Y, X)} and Y_{σ} all denote Y with endowed with ).
𝒢 ⊆ 𝒫(X) ("topology of uniform convergence on ...") 
Notation  Name ("topology of...")  Alternative name 

finite subsets of X (or closed disked hulls of finite subsets of X) 
pointwise/simple convergence  weak/weak* topology  
compact disks  Mackey topology  
compact convex subsets  compact convex convergence  
compact subsets (or balanced compact subsets) 
compact convergence  
complete and bounded disks  convex balanced complete bounded convergence  
precompact/totally bounded subsets (or balanced precompact subsets) 
precompact convergence  
infracomplete and bounded disks  convex balanced infracomplete bounded convergence  
bounded subsets  bounded convergence  strong topology Strongest polar topology 
Weak topology σ(Y, X)
For any x ∈ X, a basic neighborhood of x in X is a set of the form:
for some real r > 0 and some finite set of points y_{1}, ..., y_{n} in Y.^{[3]}
The continuous dual space of is X, where more precisely, this means that a linear functional f on Y belongs to this continuous dual space if and only if there exists some x ∈ X such that f (y) = b(x, y) for all y ∈ Y.^{[3]} The weak topology is the coarsest TVS topology on Y for which this is true.
In general, the convex balanced hull of a compact subset of Y need not be compact.^{[3]}
If X and Y are vector spaces over the complex numbers (which implies that b is complex valued) then let and denote these spaces when they are considered as vector spaces over the real numbers ℝ. Let Re b denote the real part of b and observe that (X_{ℝ}, Y_{ℝ}, Re b) is a pairing. The weak topology on Y is identical to the weak topology 𝜎(X_{ℝ}, Y_{ℝ}, Re b). This ultimately stems from the fact that for any complexvalued linear functional f on Y with real part r := Re f, then
 f = r (y)  i r (i y) for all y ∈ Y.
Mackey topology τ(Y, X)
The continuous dual space of is X (in the exact same way as was described for the weak topology). Moreover, the Mackey topology is the finest locally convex topology on Y for which this is true, which is what makes this topology important.
Since in general, the convex balanced hull of a compact subset of Y need not be compact,^{[3]} the Mackey topology may be strictly coarser than the topology Since every compact set is bounded, the Mackey topology is coarser than the strong topology .^{[3]}
Strong topology 𝛽(Y, X)
A neighborhood basis (not just a subbasis) at the origin for the topology is:^{[3]}
The strong topology is finer than the Mackey topology.^{[3]}
Polar topologies and topological vector spaces
Throughout this section, X will be a topological vector space (TVS) with continuous dual space and will be the canonical pairing, where by definition The vector space X always distinguishes/separates the points of but may fail to distinguishes the points of X (this necessarily happens if, for instance, X is not Hausdorff), in which case the pairing is not a dual pair. By the HahnBanach theorem, if X is a Hausdorff locally convex space then separates points of X and thus forms a dual pair.
Properties
 If G covers X then the canonical map from X into is welldefined. That is, for all x ∈ X the evaluation functional on meaning the map is continuous on
 If in addition separates points on X then the canonical map of X into is an injection.
 Suppose that u : E → F is a continuous linear and that 𝒢 and ℋ are collections of bounded subsets of X and Y, respectively, that each satisfy axioms 𝒢_{1} and 𝒢_{2}. Then the transpose of u, is continuous if for every G ∈ 𝒢 there is some H ∈ ℋ such that u(G) ⊆ H.^{[6]}
 In particular, the transpose of u is continuous if carries the (respectively, ) topology and carry any topology stronger than the topology (respectively, ).
 If X is a locally convex Hausdorff TVS over the field 𝕂 and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢_{1} and 𝒢_{2} then the bilinear map defined by is continuous if and only if X is normable and the 𝒢topology on is the strong dual topology .
 Suppose that X is a Fréchet space and 𝒢 is a collection of bounded subsets of X that satisfies axioms 𝒢_{1} and 𝒢_{2}. If 𝒢 contains all compact subsets of X then is complete.
Polar topologies on the continuous dual space
Throughout, X will be a TVS over the field 𝕂 with continuous dual space and X and will be associated with the canonical pairing. The table below defines many of the most common polar topologies on
 Notation: If denotes a polar topology then endowed with this topology will be denoted by (e.g. if then 𝛥 = 𝜏 and so that denotes with endowed with ).
If in addition, then this TVS may be denoted by (e.g. ).
𝒢 ⊆ 𝒫(X) ("topology of uniform convergence on ...") 
Notation  Name ("topology of...")  Alternative name 

finite subsets of X (or closed disked hulls of finite subsets of X) 
pointwise/simple convergence  weak/weak* topology  
compact convex subsets  compact convex convergence  
compact subsets (or balanced compact subsets) 
compact convergence  
compact disks  Mackey topology  
precompact/totally bounded subsets (or balanced precompact subsets) 
precompact convergence  
complete and bounded disks  convex balanced complete bounded convergence  
infracomplete and bounded disks  convex balanced infracomplete bounded convergence  
bounded subsets  bounded convergence  strong topology  
compact disks in  Mackey topology 
The reason why some of the above collections (in the same row) induce the same polar topologies is due to some basic results. A closed subset of a complete TVS is complete and that a complete subset of a Hausdorff and complete TVS is closed.^{[7]} Furthermore, in every TVS, compact subsets are complete^{[7]} and the balanced hull of a compact (resp. totally bounded) subset is again compact (resp. totally bounded).^{[8]} Also, a Banach space can be complete without being weakly complete.
If B ⊆ X is bounded then B° is absorbing in (note that being absorbing is a necessary condition for B° to be a neighborhood of the origin in any TVS topology on ).^{[2]} If X is a locally convex space and B° is absorbing in then B is bounded in X. Moreover, a subset S ⊆X is weakly bounded if and only if S° is absorbing in For this reason, it is common to restrict attention to families of bounded subsets of X.
Weak/weak* topology σ(X', X)
The topology has the following properties:
 Banach–Alaoglu theorem: Every equicontinuous subset of is relatively compact for .^{[9]}
 it follows that the closure of the convex balanced hull of an equicontinuous subset of is equicontinuous and compact.
 Theorem (S. Banach): Suppose that X and Y are Fréchet spaces or that they are duals of reflexive Fréchet spaces and that is a continuous linear map. Then u is surjective if and only if the transpose of u, is onetoone and the range of is weakly closed in .
 Suppose that X and Y are Fréchet spaces, Z is a Hausdorff locally convex space and that is a separatelycontinuous bilinear map. Then is continuous.
 In particular, any separately continuous bilinear maps from the product of two duals of reflexive Fréchet spaces into a third one is continuous.
 is normable if and only if X is finitedimensional.
 When X is infinitedimensional the topology on is strictly coarser than the strong dual topology .
 Suppose that X is a locally convex Hausdorff space and that is its completion. If then is strictly finer than .
 Any equicontinuous subset in the dual of a separable Hausdorff locally convex vector space is metrizable in the topology.
 If X is locally convex then a subset H of is bounded if and only if there exists a barrel B in X such that H ⊆ B°.^{[3]}
Compactconvex convergence γ(X', X)
If X is a Fréchet space then the topologies γ(X', X) = c(X', X).
Compact convergence c(X', X)
If X is a Fréchet space or a LFspace then is complete.
Suppose that X is a metrizable topological vector space and that If the intersection of with every equicontinuous subset of is weaklyopen, then is open in .
Precompact convergence
Banach–Alaoglu theorem: An equicontinuous subset K of has compact closure in the topology of uniform convergence on precompact sets. Furthermore, this topology on K coincides with the topology.
Mackey topology τ(X', X)
By letting 𝒢 be the set of all convex balanced weakly compact subsets of X, will have the Mackey topology on or the topology of uniform convergence on convex balanced weakly compact sets, which is denoted by and with this topology is denoted by .
Strong dual topology b(X', X)
Due to the importance of this topology, the continuous dual space of is commonly denoted simply by Consequently,
The topology has the following properties:
 If X is locally convex, then this topology is finer than all other 𝒢topologies on when considering only 𝒢's whose sets are subsets of X.
 If X is a bornological space (ex: metrizable or LFspace) then is complete.
 If X is a normed space then the strong dual topology on may be defined by the norm where .^{[10]}
 If X is a LFspace that is the inductive limit of the sequence of space (for ) then is a Fréchet space if and only if all are normable.
 If X is a Montel space then
 has the Heine–Borel property (i.e. every closed and bounded subset of is compact in )
 On bounded subsets of the strong and weak topologies coincide (and hence so do all other topologies finer than and coarser than ).
 Every weakly convergent sequence in is strongly convergent.
Mackey topology τ(X, X'')
By letting 𝒢'' be the set of all convex balanced weakly compact subsets of will have the Mackey topology on induced by or the topology of uniform convergence on convex balanced weakly compact subsets of , which is denoted by and with this topology is denoted by
 This topology is finer than and hence finer than .
Polar topologies induced by subsets of the continuous dual space
Throughout, X will be a TVS over the field 𝕂 with continuous dual space and the canonical pairing will be associated with X and The table below defines many of the most common polar topologies on X.
 Notation: If 𝛥(X, X') denotes a polar topology on X then X endowed with this topology will be denoted by or (e.g. for 𝜎(X, X') we'd have 𝛥 = 𝜎 so that and both denote X with endowed with 𝜎(X, X')).
𝒢 ⊆ 𝒫(X) ("topology of uniform convergence on ...") 
Notation  Name ("topology of...")  Alternative name 

finite subsets of (or closed disked hulls of finite subsets of ) 
𝜎(X, X') s(X, X') 
pointwise/simple convergence  weak topology 
equicontinuous subsets (or equicontinuous disks) (or weak* compact equicontinuous disks) 
equicontinuous convergence  
weak* compact disks  τ(X, X')  Mackey topology  
weak* compact convex subsets  γ(X, X')  compact convex convergence  
weak* compact subsets (or balanced weak* compact subsets) 
c(X, X')  compact convergence  
weak* bounded subsets  b(X, X') 𝛽(X, X') 
bounded convergence  strong topology 
The closure of an equicontinuous subset of is weak* compact and equicontinuous and furthermore, the convex balanced hull of an equicontinuous subset is equicontinuous.
Weak topology 𝜎(X, X')
Suppose that X and Y are Hausdorff locally convex spaces with X metrizable and that is a linear map. Then is continuous if and only if u : 𝜎(X, X') → 𝜎(Y, Y' is continuous. That is, is continuous when X and Y carry their given topologies if and only if u is continuous when X and Y carry their weak topologies.
Convergence on equicontinuous sets 𝜀(X, X')
If was the set of all convex balanced weakly compact equicontinuous subsets of then the same topology would have been induced.
If X is locally convex and Hausdorff then X's given topology (i.e. the topology that X started with) is exactly . That is, for X Hausdorff and locally convex, if then E is equicontinuous if and only if E° is equicontinuous and furthermore, for any S ⊆ X, S is a neighborhood of 0 if and only if S° is equicontinuous.
Importantly, a set of continuous linear functionals H on a TVS X is equicontinuous if and only if it is contained in the polar of some neighborhood U of 0 in X (i.e. H ⊆ U°). Since a TVS's topology is completely determined by the open neighborhoods of the origin, this means that via operation of taking the polar of a set, the collection of equicontinuous subsets of "encode" all information about X's topology (i.e. distinct TVS topologies on X produce distinct collections of equicontinuous subsets, and given any such collection one may recover the TVS original topology by taking the polars of sets in the collection). Thus uniform convergence on the collection of equicontinuous subsets is essentially "convergence on the topology of X".
Mackey topology τ(X, X')
Suppose that X is a locally convex Hausdorff space. If X is metrizable or barrelled then X's original topology is identical to the Mackey topology τ(X, X').^{[11]}
Topologies compatible with pairings
Let X be a vector space and let Y be a vector subspace of the algebraic dual of X that separates points on X. If 𝜏 is any other locally convex Hausdorff topological vector space topology on X, then 𝜏 is compatible with duality between X and Y if when X is equipped with 𝜏, then it has Y as its continuous dual space. If X is given the weak topology then X_{𝜎(X, Y)} is a Hausdorff locally convex topological vector space (TVS) and is compatible with duality between X and Y (i.e. ). The question arises: what are all of the locally convex Hausdorff TVS topologies that can be placed on X that are compatible with duality between X and Y? The answer to this question is called the Mackey–Arens theorem.
See also
 Dual topology
 List of topologies
 Locally convex topological vector space – A vector space with a topology defined by convex open sets
 Polar set – Subset of all points that is bounded by some given point of a dual (in a dual pairing)
 Topologies on spaces of linear maps
 Topology consistent with the duality
 Topological vector space – Vector space with a notion of nearness
Notes
 ^ Trèves 2006, p. 195.
 ^ ^{a} ^{b} ^{c} Trèves 2006, pp. 195201.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} ^{f} ^{g} ^{h} ^{i} ^{j} ^{k} ^{l} ^{m} ^{n} ^{o} ^{p} ^{q} ^{r} Narici & Beckenstein 2011, pp. 225273.
 ^ Robertson & Robertson 1964, III.2
 ^ Schaefer & Wolff 1999, p. 80.
 ^ Trèves 2006, pp. 199–200.
 ^ ^{a} ^{b} Narici & Beckenstein 2011, pp. 4766.
 ^ Narici & Beckenstein 2011, pp. 67113.
 ^ Schaefer & Wolff 1999, p. 85.
 ^ Trèves 2006, p. 198.
 ^ Trèves 2006, pp. 433.
References
 Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 9781584888666. OCLC 144216834.
 Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.
 Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 9781461271550. OCLC 840278135.
 Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 9780486453521. OCLC 853623322.