KNOWPIA
WELCOME TO KNOWPIA

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 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.

A subset in is called

- Convex if for all and In other words, contains all line segments between points in
- 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
- 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 (in the TVS , ball centered at the origin of radius , belongs, , does not belong,
*C*is not a cone but*C*is balanced,*sx*is in*C*, for all*x*belonging to*C*and scalar*s*for which ). - A cone (when the underlying field is ordered) if for all and
- 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]}

- In any TVS, every neighborhood of the origin is absorbent.
- 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, ) has 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

- is nonnegative or positive semidefinite: ;
- is positive homogeneous or positive scalable: for every scalar So, in particular, ;
- 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. By definition, it is the coarsest topology on for which all maps in are continuous. It will not carry the structure of a topological vector space (TVS) though.

The vector space operations fail to be continuous in this topology, as all sets in the semi-norm topology will be symmetric, where as the lcs topology allows for more sets.

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).

Let denote the open ball of radius in . The family of sets 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).

**Theorem ^{[11]}** — Let be a Fréchet space over the field
Then the following are equivalent:

- does
*not*admit a continuous norm (that is, any continuous seminorm on can*not*be a norm). - contains a vector subspace that is TVS-isomorphic to
- contains a complemented vector subspace that is TVS-isomorphic to

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

**Theorem ^{[7]}** — Suppose that is a (real or complex) vector space and let be a filter base of subsets of such that:

Then is a neighborhood base at 0 for a locally convex TVS topology on

**Theorem ^{[7]}** — Suppose that is a (real or complex) vector space and let be a non-empty collection of convex, balanced, and absorbing subsets of
Then the set of all positive scalar multiples of finite intersections of sets in forms a neighborhood base at the origin for a locally convex TVS topology on

**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
- 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 - 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:

**Theorem ^{[13]}** (Kalton) — Every complete metrizable TVS with the Hahn-Banach extension property is locally convex.

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]}

- 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.
- 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.

- This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is
- 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

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
- 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):

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.

**Theorem ^{[40]}** — Let be a linear operator between TVSs where is locally convex (note that need

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.

**Theorem ^{[40]}** — If is a TVS (not necessarily locally convex) and if is a linear functional on , then is continuous if and only if there exists a continuous seminorm on such that

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:

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

- Convex metric space – metric space with the property any segment joining two points in that space has other points in it besides the endpoints
- Krein–Milman theorem – On when a space equals the closed convex hull of its extreme points
- Linear form – Linear map from a vector space to its field of scalars
- Locally convex vector lattice
- Minkowski functional – Function made from a set
- Seminorm – nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenous
- Sublinear functional – Type of function in linear algebra
- Topological group – Group that is a topological space with continuous group action
- Topological vector space – Vector space with a notion of nearness
- Vector space – Algebraic structure in linear algebra

**^**Hausdorff, F.*Grundzüge der Mengenlehre*(1914)**^**von Neumann, J.*Collected works*. Vol II. pp. 94–104**^**Dieudonne, J.*History of Functional Analysis*Chapter VIII. Section 1.**^**von Neumann, J.*Collected works*. Vol II. pp. 508–527**^**Dieudonne, J.*History of Functional Analysis*Chapter VIII. Section 2.**^**Banach, S.*Theory of linear operations*p. 75. Ch. VIII. Sec. 3. Theorem 4., translated from*Theorie des operations lineaires*(1932)- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}^{h}Narici & Beckenstein 2011, pp. 67–113. **^**Narici & Beckenstein 2011, p. 83.- ^
^{a}^{b}^{c}^{d}^{e}^{f}Narici & Beckenstein 2011, p. 122. **^**Jarchow 1981, p. 130.**^**Jarchow 1981, pp. 129–130.- ^
^{a}^{b}Narici & Beckenstein 2011, p. 126. - ^
^{a}^{b}^{c}^{d}Narici & Beckenstein 2011, pp. 225–273. **^**Narici & Beckenstein 2011, p. 149.- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}Narici & Beckenstein 2011, pp. 149–153. **^**Narici & Beckenstein 2011, pp. 115–154.**^**Bessaga & Pełczyński 1975, p. 189**^**Rudin 1991, p. 6.**^**Rudin 1991, p. 38.- ^
^{a}^{b}^{c}^{d}^{e}^{f}Trèves 2006, p. 126. **^**Narici & Beckenstein 2011, pp. 177–220.- ^
^{a}^{b}Schaefer & Wolff 1999, p. 38. **^**Jarchow 1981, pp. 101–104.**^**Conway 1990, p. 102.**^**Trèves 2006, p. 370.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 155–176. **^**Rudin 1991, p. 7.**^**Aliprantis & Border 2006, p. 185.**^**Trèves 2006, p. 67.**^**Trèves 2006, p. 145.- ^
^{a}^{b}Rudin 1991, pp. 72–73. **^**Trèves 2006, p. 362.- ^
^{a}^{b}Trèves 2006, p. 68. - ^
^{a}^{b}Narici & Beckenstein 2011, p. 108. - ^
^{a}^{b}Dunford 1988, p. 415. **^**Rudin 1991, pp. 73–74.- ^
^{a}^{b}^{c}Narici & Beckenstein 2011, pp. 125–126. **^**Narici & Beckenstein 2011, p. 476.**^**Narici & Beckenstein 2011, p. 446.- ^
^{a}^{b}Narici & Beckenstein 2011, pp. 126–128. **^**Narici & Beckenstein 2011, pp. 126-–128.

**^**Let be the open unit ball associated with the seminorm and note that if is real then and so Thus a basic open neighborhood of the origin induced by is a finite intersection of the form where and are all positive reals. Let which is a continuous seminorm and moreover, Pick and such that where this inequality holds if and only if Thus as desired.**^**Fix so it remains to show that belongs to By replacing with if necessary, we may assume without loss of generality that and so it remains to show that is a neighborhood of the origin. Let so that Since scalar multiplication by is a linear homeomorphism Since and it follows that where because is open, there exists some which satisfies Define by which is a homeomorphism because The set is thus an open subset of that moreover contains If then since is convex, and which proves that Thus is an open subset of that contains the origin and is contained in Q.E.D.

- Aliprantis, Charalambos D.; Border, Kim C. (2006).
*Infinite Dimensional Analysis: A Hitchhiker's Guide*(Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874. - Berberian, Sterling K. (1974).
*Lectures in Functional Analysis and Operator Theory*. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. - Bessaga, C.; Pełczyński, A. (1975),
*Selected Topics in Infinite-Dimensional Topology*, Monografie Matematyczne, Warszawa: Panstwowe wyd. naukowe. - Bourbaki, Nicolas (1987) [1981].
*Topological Vector Spaces: Chapters 1–5*. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. - Conway, John (1990).
*A course in functional analysis*. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. - Dunford, Nelson (1988).
*Linear operators*(in Romanian). New York: Interscience Publishers. ISBN 0-471-60848-3. OCLC 18412261. - Edwards, Robert E. (1995).
*Functional Analysis: Theory and Applications*. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. - Grothendieck, Alexander (1973).
*Topological Vector Spaces*. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. - 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. - Rudin, Walter (1991).
*Functional Analysis*. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. - Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Swartz, Charles (1992).
*An introduction to Functional Analysis*. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. - 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.