In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively they can be defined as a vector space with a family of seminorms, and a topology can be defined in terms of that family. Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn–Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Fréchet spaces are locally convex topological vector spaces that are completely metrizable (with a choice of complete metric). They are generalizations of Banach spaces, which are complete vector spaces with respect to a metric generated by a norm.

Metrizable topologies on vector spaces have been studied since their introduction in Maurice Fréchet's 1902 PhD thesis Sur quelques points du calcul fonctionnel (wherein the notion of a metric was first introduced). After the notion of a general topological space was defined by Felix Hausdorff in 1914,^[1] although locally convex topologies were implicitly used by some mathematicians, up to 1934 only John von Neumann would seem to have explicitly defined the weak topology on Hilbert spaces and strong operator topology on operators on Hilbert spaces.^[2][3] Finally, in 1935 von Neumann introduced the general definition of a locally convex space (called a convex space by him).^[4][5]

A notable example of a result which had to wait for the development and dissemination of general locally convex spaces (amongst other notions and results, like nets, the product topology and Tychonoff's theorem) to be proven in its full generality, is the Banach–Alaoglu theorem which Stefan Banach first established in 1932 by an elementary diagonal argument for the case of separable normed spaces^[6] (in which case the unit ball of the dual is metrizable).

Suppose is a vector space over a subfield of the complex numbers (normally itself or ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

A topological vector space (TVS) is called locally convex if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets.^[7] The term locally convex topological vector space is sometimes shortened to locally convex space or LCTVS.

A subset in is called

1. Convex if for all and In other words, contains all line segments between points in
2. Circled if for all and scalars if then If this means that is equal to its reflection through the origin. For it means for any contains the circle through centred on the origin, in the one-dimensional complex subspace generated by
3. Balanced if for all and scalars if then If this means that if then contains the line segment between and For it means for any contains the disk with on its boundary, centred on the origin, in the one-dimensional complex subspace generated by Equivalently, a balanced set is a "circled cone"^{[citation needed]}. Note that in the TVS , belongs to ball centered at the origin of radius , but does not belong; indeed, C is not a cone, but is balanced.
4. A cone (when the underlying field is ordered) if for all and
5. Absorbent or absorbing if for every there exists such that for all satisfying The set can be scaled out by any "large" value to absorb every point in the space.
   • In any TVS, every neighborhood of the origin is absorbent.^[7]
6. Absolutely convex or a disk if it is both balanced and convex. This is equivalent to it being closed under linear combinations whose coefficients absolutely sum to ; such a set is absorbent if it spans all of

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets.^[8] Every TVS has a neighborhood basis at the origin consisting of balanced sets, but only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both balanced and convex. It is possible for a TVS to have some neighborhoods of the origin that are convex and yet not be locally convex because it has no neighborhood basis at the origin consisting entirely of convex sets (that is, every neighborhood basis at the origin contains some non-convex set); for example, every non-locally convex TVS has itself (that is, ) as a convex neighborhood of the origin.

Because translation is continuous (by definition of topological vector space), all translations are homeomorphisms, so every base for the neighborhoods of the origin can be translated to a base for the neighborhoods of any given vector.

A seminorm on is a map such that

1. is nonnegative or positive semidefinite: ;
2. is positive homogeneous or positive scalable: for every scalar So, in particular, ;
3. is subadditive. It satisfies the triangle inequality:

If satisfies positive definiteness, which states that if then then is a norm. While in general seminorms need not be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

If is a vector space and is a family of seminorms on then a subset of is called a base of seminorms for if for all there exists a and a real such that ^[9]

Definition (second version): A locally convex space is defined to be a vector space along with a family of seminorms on

Suppose that is a vector space over where is either the real or complex numbers. A family of seminorms on the vector space induces a canonical vector space topology on , called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on for which all maps in are continuous.

It is possible for a locally convex topology on a space to be induced by a family of norms but for to not be normable (that is, to have its topology be induced by a single norm).

An open set in has the form , where is a positive real number. The family of preimages as ranges over a family of seminorms and ranges over the positive real numbers is a subbasis at the origin for the topology induced by . These sets are convex, as follows from properties 2 and 3 of seminorms. Intersections of finitely many such sets are then also convex, and since the collection of all such finite intersections is a basis at the origin it follows that the topology is locally convex in the sense of the first definition given above.

Recall that the topology of a TVS is translation invariant, meaning that if is any subset of containing the origin then for any is a neighborhood of the origin if and only if is a neighborhood of ; thus it suffices to define the topology at the origin. A base of neighborhoods of for this topology is obtained in the following way: for every finite subset of and every let

If is a locally convex space and if is a collection of continuous seminorms on , then is called a base of continuous seminorms if it is a base of seminorms for the collection of all continuous seminorms on .^[9] Explicitly, this means that for all continuous seminorms on , there exists a and a real such that ^[9] If is a base of continuous seminorms for a locally convex TVS then the family of all sets of the form as varies over and varies over the positive real numbers, is a base of neighborhoods of the origin in (not just a subbasis, so there is no need to take finite intersections of such sets).^{[9][proof 1]}

A family of seminorms on a vector space is called saturated if for any and in the seminorm defined by belongs to

If is a saturated family of continuous seminorms that induces the topology on then the collection of all sets of the form as ranges over and ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open sets;^[9] This forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to take finite intersections of such sets.^[9]

The following theorem implies that if is a locally convex space then the topology of can be a defined by a family of continuous norms on (a norm is a seminorm where implies ) if and only if there exists at least one continuous norm on .^[10] This is because the sum of a norm and a seminorm is a norm so if a locally convex space is defined by some family of seminorms (each of which is necessarily continuous) then the family of (also continuous) norms obtained by adding some given continuous norm to each element, will necessarily be a family of norms that defines this same locally convex topology. If there exists a continuous norm on a topological vector space then is necessarily Hausdorff but the converse is not in general true (not even for locally convex spaces or Fréchet spaces).

Suppose that the topology of a locally convex space is induced by a family of continuous seminorms on . If and if is a net in , then in if and only if for all ^[12] Moreover, if is Cauchy in , then so is for every ^[12]

Although the definition in terms of a neighborhood base gives a better geometric picture, the definition in terms of seminorms is easier to work with in practice. The equivalence of the two definitions follows from a construction known as the Minkowski functional or Minkowski gauge. The key feature of seminorms which ensures the convexity of their -balls is the triangle inequality.

For an absorbing set such that if then whenever define the Minkowski functional of to be

From this definition it follows that is a seminorm if is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets form a base of convex absorbent balanced sets.

Example: auxiliary normed spaces

If is convex and absorbing in then the symmetric set will be convex and balanced (also known as an absolutely convex set or a disk) in addition to being absorbing in This guarantees that the Minkowski functional of will be a seminorm on thereby making into a seminormed space that carries its canonical pseudometrizable topology. The set of scalar multiples as ranges over (or over any other set of non-zero scalars having as a limit point) forms a neighborhood basis of absorbing disks at the origin for this locally convex topology. If is a topological vector space and if this convex absorbing subset is also a bounded subset of then the absorbing disk will also be bounded, in which case will be a norm and will form what is known as an auxiliary normed space. If this normed space is a Banach space then is called a Banach disk.

• A family of seminorms is called total or separated or is said to separate points if whenever holds for every then is necessarily A locally convex space is Hausdorff if and only if it has a separated family of seminorms. Many authors take the Hausdorff criterion in the definition.
• A pseudometric is a generalization of a metric which does not satisfy the condition that only when A locally convex space is pseudometrizable, meaning that its topology arises from a pseudometric, if and only if it has a countable family of seminorms. Indeed, a pseudometric inducing the same topology is then given by (where the can be replaced by any positive summable sequence ). This pseudometric is translation-invariant, but not homogeneous, meaning and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and only if the family of seminorms is separated, since this is the case if and only if the space is Hausdorff. If furthermore the space is complete, the space is called a Fréchet space.
• As with any topological vector space, a locally convex space is also a uniform space. Thus one may speak of uniform continuity, uniform convergence, and Cauchy sequences.
• A Cauchy net in a locally convex space is a net such that for every and every seminorm there exists some index such that for all indices In other words, the net must be Cauchy in all the seminorms simultaneously. The definition of completeness is given here in terms of nets instead of the more familiar sequences because unlike Fréchet spaces which are metrizable, general spaces may be defined by an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to characterize convergence in such spaces. A locally convex space is complete if and only if every Cauchy net converges.
• A family of seminorms becomes a preordered set under the relation if and only if there exists an such that for all One says it is a directed family of seminorms if the family is a directed set with addition as the join, in other words if for every and there is a such that Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family let be the set of finite subsets of and then for every define One may check that is an equivalent directed family.
• If the topology of the space is induced from a single seminorm, then the space is seminormable. Any locally convex space with a finite family of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given by the sum of the seminorms. In terms of the open sets, a locally convex topological vector space is seminormable if and only if the origin has a bounded neighborhood.

Let be a TVS. Say that a vector subspace of has the extension property if any continuous linear functional on can be extended to a continuous linear functional on .^[13] Say that has the Hahn-Banach extension property (HBEP) if every vector subspace of has the extension property.^[13]

The Hahn-Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable TVSs there is a converse:

If a vector space has uncountable dimension and if we endow it with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.^[13]

Throughout, is a family of continuous seminorms that generate the topology of

Topological closure

If and then if and only if for every and every finite collection there exists some such that ^[14] The closure of in is equal to ^[15]

Topology of Hausdorff locally convex spaces

Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces.^[16] The Anderson–Kadec theorem states that every infinite–dimensional separable Fréchet space is homeomorphic to the product space of countably many copies of (this homeomorphism need not be a linear map).^[17]

Algebraic properties of convex subsets

A subset is convex if and only if for all ^[18] or equivalently, if and only if for all positive real ^[19] where because always holds, the equals sign can be replaced with If is a convex set that contains the origin then is star shaped at the origin and for all non-negative real

The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.^[20]

Topological properties of convex subsets

• Suppose that is a TVS (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of are exactly those that are of the form for some and some positive continuous sublinear functional on ^[21]
• The interior and closure of a convex subset of a TVS is again convex.^[20]
• If is a convex set with non-empty interior, then the closure of is equal to the closure of the interior of ; furthermore, the interior of is equal to the interior of the closure of ^[20][22]
  • So if the interior of a convex set is non-empty then is a closed (respectively, open) set if and only if it is a regular closed (respectively, regular open) set.
• If is convex and then^[23] Explicitly, this means that if is a convex subset of a TVS (not necessarily Hausdorff or locally convex), belongs to the closure of and belongs to the interior of then the open line segment joining and belongs to the interior of that is, ^{[22][24][proof 2]}
• If is a closed vector subspace of a (not necessarily Hausdorff) locally convex space is a convex neighborhood of the origin in and if is a vector not in then there exists a convex neighborhood of the origin in such that and ^[20]
• The closure of a convex subset of a locally convex Hausdorff space is the same for all locally convex Hausdorff TVS topologies on that are compatible with duality between and its continuous dual space.^[25]
• In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.^[7]
• In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.^[7]
  • More generally, if is a compact subset of a locally convex space, then the convex hull (respectively, the disked hull ) is compact if and only if it is complete.^[7]
• In a locally convex space, convex hulls of bounded sets are bounded. This is not true for TVSs in general.^[26]
• In a Fréchet space, the closed convex hull of a compact set is compact.^[27]
• In a locally convex space, any linear combination of totally bounded sets is totally bounded.^[26]

For any subset of a TVS the convex hull (respectively, closed convex hull, balanced hull, convex balanced hull) of denoted by (respectively, ), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of containing

• The convex hull of compact subset of a Hilbert space is not necessarily closed and so also not necessarily compact. For example, let be the separable Hilbert space of square-summable sequences with the usual norm and let be the standard orthonormal basis (that is at the -coordinate). The closed set is compact but its convex hull is not a closed set because belongs to the closure of in but (since every sequence is a finite convex combination of elements of and so is necessarily in all but finitely many coordinates, which is not true of ).^[28] However, like in all complete Hausdorff locally convex spaces, the closed convex hull of this compact subset is compact. The vector subspace is a pre-Hilbert space when endowed with the substructure that the Hilbert space induces on it but is not complete and (since ). The closed convex hull of in (here, "closed" means with respect to and not to as before) is equal to which is not compact (because it is not a complete subset). This shows that in a Hausdorff locally convex space that is not complete, the closed convex hull of compact subset might fail to be compact (although it will be precompact/totally bounded).
• In a Hausdorff locally convex space the closed convex hull of compact subset is not necessarily compact although it is a precompact (also called "totally bounded") subset, which means that its closure, when taken in a completion of will be compact (here so that if and only if is complete); that is to say, will be compact. So for example, the closed convex hull of a compact subset of of a pre-Hilbert space is always a precompact subset of and so the closure of in any Hilbert space containing (such as the Hausdorff completion of for instance) will be compact (this is the case in the previous example above).
• In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is again compact.
• In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact.^[29] Consequently, in a complete Hausdorff locally convex space, the closed convex hull of a compact subset is again compact.^[30]
• In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).^[7]
  • This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is closed (in addition to being compact^[31] and convex); in particular, the convex hull of such a union is equal to the closed convex hull of that union.
  • In general, the closed convex hull of a compact set is not necessarily compact. However, every compact subset of (where ) does have a compact convex hull.^[31]
  • In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but not closed.
• The bipolar theorem states that the bipolar (that is, the polar of the polar) of a subset of a locally convex Hausdorff TVS is equal to the closed convex balanced hull of that set.^[32]
• The balanced hull of a convex set is not necessarily convex.
• If and are convex subsets of a topological vector space and if then there exist and a real number satisfying such that ^[20]
• If is a vector subspace of a TVS a convex subset of and a convex subset of such that then ^[20]
• Recall that the smallest balanced subset of containing a set is called the balanced hull of and is denoted by For any subset of the convex balanced hull of denoted by is the smallest subset of containing that is convex and balanced.^[33] The convex balanced hull of is equal to the convex hull of the balanced hull of (i.e. ), but the convex balanced hull of is not necessarily equal to the balanced hull of the convex hull of (that is, is not necessarily equal to ).^[33]
• If are subsets of a TVS and if is a scalar then ^[34] and Moreover, if is compact then ^[35] However, the convex hull of a closed set need not be closed;^[34] for example, the set is closed in but its convex hull is the open set
• If are subsets of a TVS whose closed convex hulls are compact, then ^[35]
• If is a convex set in a complex vector space and there exists some such that then for all real such that In particular, for all scalars such that
• Carathéodory's theorem: If is any subset of (where ) then for every there exist a finite subset containing at most points whose convex hull contains (that is, and ).^[36]

Any vector space endowed with the trivial topology (also called the indiscrete topology) is a locally convex TVS (and of course, it is the coarsest such topology). This topology is Hausdorff if and only The indiscrete topology makes any vector space into a complete pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on if and only This follows from the fact that every topological vector space is a connected space.

If is a real or complex vector space and if is the set of all seminorms on then the locally convex TVS topology, denoted by that induces on is called the finest locally convex topology on ^[37] This topology may also be described as the TVS-topology on having as a neighborhood base at the origin the set of all absorbing disks in ^[37] Any locally convex TVS-topology on is necessarily a subset of is Hausdorff.^[15] Every linear map from into another locally convex TVS is necessarily continuous.^[15] In particular, every linear functional on is continuous and every vector subspace of is closed in ;^[15] therefore, if is infinite dimensional then is not pseudometrizable (and thus not metrizable).^[37] Moreover, is the only Hausdorff locally convex topology on with the property that any linear map from it into any Hausdorff locally convex space is continuous.^[38] The space is a bornological space.^[39]

Every normed space is a Hausdorff locally convex space, and much of the theory of locally convex spaces generalizes parts of the theory of normed spaces. The family of seminorms can be taken to be the single norm. Every Banach space is a complete Hausdorff locally convex space, in particular, the spaces with are locally convex.

More generally, every Fréchet space is locally convex. A Fréchet space can be defined as a complete locally convex space with a separated countable family of seminorms.

The space of real valued sequences with the family of seminorms given by is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is not normable. This is also the limit topology of the spaces embedded in in the natural way, by completing finite sequences with infinitely many

Given any vector space and a collection of linear functionals on it, can be made into a locally convex topological vector space by giving it the weakest topology making all linear functionals in continuous. This is known as the weak topology or the initial topology determined by The collection may be the algebraic dual of or any other collection. The family of seminorms in this case is given by for all in

Spaces of differentiable functions give other non-normable examples. Consider the space of smooth functions such that where and are multiindices. The family of seminorms defined by is separated, and countable, and the space is complete, so this metrizable space is a Fréchet space. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the space of tempered distributions.

An important function space in functional analysis is the space of smooth functions with compact support in A more detailed construction is needed for the topology of this space because the space is not complete in the uniform norm. The topology on is defined as follows: for any fixed compact set the space of functions with is a Fréchet space with countable family of seminorms (these are actually norms, and the completion of the space with the norm is a Banach space ). Given any collection of compact sets, directed by inclusion and such that their union equal the form a direct system, and is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More concretely, is the union of all the with the strongest locally convex topology which makes each inclusion map continuous. This space is locally convex and complete. However, it is not metrizable, and so it is not a Fréchet space. The dual space of is the space of distributions on

More abstractly, given a topological space the space of continuous (not necessarily bounded) functions on can be given the topology of uniform convergence on compact sets. This topology is defined by semi-norms (as varies over the directed set of all compact subsets of ). When is locally compact (for example, an open set in ) the Stone–Weierstrass theorem applies—in the case of real-valued functions, any subalgebra of that separates points and contains the constant functions (for example, the subalgebra of polynomials) is dense.

Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:

• The spaces for are equipped with the F-norm They are not locally convex, since the only convex neighborhood of zero is the whole space. More generally the spaces with an atomless, finite measure and are not locally convex.
• The space of measurable functions on the unit interval (where we identify two functions that are equal almost everywhere) has a vector-space topology defined by the translation-invariant metric (which induces the convergence in measure of measurable functions; for random variables, convergence in measure is convergence in probability): This space is often denoted

Both examples have the property that any continuous linear map to the real numbers is In particular, their dual space is trivial, that is, it contains only the zero functional.

• The sequence space is not locally convex.

Because locally convex spaces are topological spaces as well as vector spaces, the natural functions to consider between two locally convex spaces are continuous linear maps. Using the seminorms, a necessary and sufficient criterion for the continuity of a linear map can be given that closely resembles the more familiar boundedness condition found for Banach spaces.

Given locally convex spaces and with families of seminorms and respectively, a linear map is continuous if and only if for every there exist and such that for all

In other words, each seminorm of the range of is bounded above by some finite sum of seminorms in the domain. If the family is a directed family, and it can always be chosen to be directed as explained above, then the formula becomes even simpler and more familiar:

The class of all locally convex topological vector spaces forms a category with continuous linear maps as morphisms.

If is a real or complex vector space, is a linear functional on , and is a seminorm on , then if and only if ^[41] If is a non-0 linear functional on a real vector space and if is a seminorm on , then if and only if ^[15]

Let be an integer, be TVSs (not necessarily locally convex), let be a locally convex TVS whose topology is determined by a family of continuous seminorms, and let be a multilinear operator that is linear in each of its coordinates. The following are equivalent:

1. is continuous.
2. For every there exist continuous seminorms on respectively, such that for all ^[15]
3. For every there exists some neighborhood of the origin in on which is bounded.^[15]

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