Generating Family of Seminorms in a Topological Vector Space

A vector space with a topology divers by convex open sets

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 divers as topological vector spaces whose topology is generated past translations of balanced, absorbent, convex sets. Alternatively they can be defined equally a vector space with a family unit of seminorms, and a topology can be defined in terms of that family unit. Although in general such spaces are not necessarily normable, the being of a convex local base for the zip vector is strong enough for the Hahn–Banach theorem to concord, 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.

History [edit]

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 commencement introduced). After the notion of a general topological space was divers 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 accept explicitly defined the weak topology on Hilbert spaces and stiff 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 past him).^{[four] [five]}

A notable example of a result which had to wait for the development and dissemination of full general locally convex spaces (amongst other notions and results, similar nets, the product topology and Tychonoff's theorem) to exist proven in its total 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^[six] (in which case the unit ball of the dual is metrizable).

Definition [edit]

Suppose ${\displaystyle X}$ is a vector space over ${\displaystyle \mathbb {Grand} ,}$ a subfield of the complex numbers (normally ${\displaystyle \mathbb {C} }$ itself or ${\displaystyle \mathbb {R} }$ ). A locally convex space is defined either in terms of convex sets, or equivalently in terms of seminorms.

Definition via convex sets [edit]

A subset ${\displaystyle C}$ in ${\displaystyle X}$ is called

1. Convex if for all ${\displaystyle ten,y\in C,}$ and ${\displaystyle 0\leq t\leq ane,}$ ${\displaystyle tx+(1-t)y\in C.}$ In other words, ${\displaystyle C}$ contains all line segments betwixt points in ${\displaystyle C.}$
2. Circled if for all ${\displaystyle 10\in C}$ and scalars ${\displaystyle s,}$ if ${\displaystyle |due south|=1}$ and then ${\displaystyle sx\in C.}$ If ${\displaystyle \mathbb {One thousand} =\mathbb {R} ,}$ this ways that ${\displaystyle C}$ is equal to its reflection through the origin. For ${\displaystyle \mathbb {K} =\mathbb {C} ,}$ it means for any ${\displaystyle ten\in C,}$ ${\displaystyle C}$ contains the circle through ${\displaystyle x,}$ centred on the origin, in the one-dimensional complex subspace generated by ${\displaystyle ten.}$
3. Counterbalanced if for all ${\displaystyle ten\in C}$ and scalars ${\displaystyle s,}$ if ${\displaystyle |s|\leq 1}$ so ${\displaystyle sx\in C.}$ If ${\displaystyle \mathbb {1000} =\mathbb {R} ,}$ this ways that if ${\displaystyle x\in C,}$ then ${\displaystyle C}$ contains the line segment between ${\displaystyle x}$ and ${\displaystyle -ten.}$ For ${\displaystyle \mathbb {Grand} =\mathbb {C} ,}$ it ways for any in ${\displaystyle x\in C,}$ ${\displaystyle C}$ contains the disk with ${\displaystyle ten}$ on its boundary, centred on the origin, in the one-dimensional complex subspace generated by ${\displaystyle ten.}$ Equivalently, a balanced set is a circled cone.
4. A cone (when the underlying field is ordered) if for all ${\displaystyle x\in C}$ and ${\displaystyle 0\leq t,}$ ${\displaystyle tx\in C.}$
5. Absorbent or absorbing if for every ${\displaystyle x\in X,}$ there exists ${\displaystyle r>0}$ such that ${\displaystyle x\in tC}$ for all ${\displaystyle t\in \mathbb {K} }$ satisfying ${\displaystyle |t|>r.}$ The prepare ${\displaystyle C}$ can be scaled out by any "big" value to blot every point in the infinite.
   • In any TVS, every neighborhood of the origin is absorbent.^[7]
6. Admittedly convex or a deejay if it is both balanced and convex. This is equivalent to information technology being closed under linear combinations whose coefficients absolutely sum to ${\displaystyle \leq i}$ ; such a set is absorbent if it spans all of ${\displaystyle X.}$

A topological vector space (TVS) is called locally convex if the origin has a neighborhood ground (that is, a local base) consisting of convex sets.^[vii]

In fact, every locally convex TVS has a neighborhood basis of the origin consisting of absolutely convex sets (that is, disks), where this neighborhood ground tin farther be chosen to also consist entirely of open sets or entirely of airtight sets.^[seven] Every TVS has a neighborhood footing at the origin consisting of balanced sets only only a locally convex TVS has a neighborhood basis at the origin consisting of sets that are both counterbalanced and convex. It is possible for a TVS to take some neighborhoods of the origin that are convex and yet non be locally convex.

Considering translation is (by definition of "topological vector space") continuous, 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.

Definition via seminorms [edit]

A seminorm on ${\displaystyle X}$ is a map ${\displaystyle p:Ten\to \mathbb {R} }$ such that

1. ${\displaystyle p}$ is positive or positive semidefinite: ${\displaystyle p(x)\geq 0}$ ;
2. ${\displaystyle p}$ is positive homogeneous or positive scalable: ${\displaystyle p(sx)=|due south|p(x)}$ for every scalar ${\displaystyle s.}$ So, in particular, ${\displaystyle p(0)=0}$ ;
3. ${\displaystyle p}$ is subadditive. It satisfies the triangle inequality: ${\displaystyle p(x+y)\leq p(x)+p(y).}$

If ${\displaystyle p}$ satisfies positive definiteness, which states that if ${\displaystyle p(x)=0}$ then ${\displaystyle x=0,}$ and so ${\displaystyle p}$ is a norm. While in general seminorms need non be norms, there is an analogue of this criterion for families of seminorms, separatedness, defined below.

If ${\displaystyle X}$ is a vector infinite and ${\displaystyle {\mathcal {P}}}$ is a family of seminorms on ${\displaystyle X}$ then a subset ${\displaystyle {\mathcal {Q}}}$ of ${\displaystyle {\mathcal {P}}}$ is called a base of seminorms for ${\displaystyle {\mathcal {P}}}$ if for all ${\displaystyle p\in {\mathcal {P}}}$ there exists a ${\displaystyle q\in {\mathcal {Q}}}$ and a real ${\displaystyle r>0}$ such that ${\displaystyle p\leq rq.}$ ^[8]

Definition (second version): A locally convex infinite is defined to be a vector space ${\displaystyle X}$ forth with a family ${\displaystyle {\mathcal {P}}}$ of seminorms on ${\displaystyle Ten.}$

Seminorm topology [edit]

Suppose that ${\displaystyle Ten}$ is a vector space over ${\displaystyle \mathbb {K} ,}$ where ${\displaystyle \mathbb {K} }$ is either the existent or complex numbers, and allow ${\displaystyle B_{r}}$ (resp. ${\displaystyle B_{\leq r}}$ ) denote the open (resp. closed) ball of radius ${\displaystyle r>0}$ in ${\displaystyle \mathbb {K} .}$ A family of seminorms ${\displaystyle {\mathcal {P}}}$ on the vector space ${\displaystyle X}$ induces a canonical vector space topology on ${\displaystyle X}$ , called the initial topology induced by the seminorms, making it into a topological vector space (TVS). By definition, it is the coarsest topology on ${\displaystyle Ten}$ for which all maps in ${\displaystyle {\mathcal {P}}}$ are continuous.

That the vector space operations are continuous in this topology follows from properties 2 and three above. Information technology tin easily exist seen that the resulting topological vector space is "locally convex" in the sense of the kickoff definition given above because each ${\displaystyle U_{B,r}(0)}$ is absolutely convex and absorptive (and because the latter backdrop are preserved by translations).

Information technology is possible for a locally convex topology on a infinite ${\displaystyle X}$ to be induced by a family of norms but for ${\displaystyle 10}$ to not exist normable (that is, to have its topology be induced by a single norm).

Basis and subbasis [edit]

Suppose that ${\displaystyle {\mathcal {P}}}$ is a family of seminorms on ${\displaystyle 10}$ that induces a locally convex topology ${\displaystyle \tau }$ on ${\displaystyle X}$ . A subbasis at the origin is given past all sets of the course ${\displaystyle p^{-1}\left(B_{<r}\right)=\{x\in Ten:p(x)<r\}}$ equally ${\displaystyle p}$ ranges over ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle r}$ ranges over the positive existent numbers. A base at the origin is given by the drove of all possible finite intersections of such subbasis sets.

Think that the topology of a TVS is translation invariant, meaning that if ${\displaystyle S}$ is any subset of ${\displaystyle X}$ containing the origin then for whatsoever ${\displaystyle x\in X,}$ ${\displaystyle S}$ is a neighborhood of the origin if and only if ${\displaystyle x+S}$ is a neighborhood of ${\displaystyle x}$ ; thus it suffices to ascertain the topology at the origin. A base of neighborhoods of ${\displaystyle y}$ for this topology is obtained in the following fashion: for every finite subset ${\displaystyle F}$ of ${\displaystyle {\mathcal {P}}}$ and every ${\displaystyle r>0,}$ let

$${\displaystyle U_{F,r}(y):=\{x\in X:p(x-y)<r\ {\text{ for all }}p\in F\}.}$$

Bases of seminorms and saturated families [edit]

If ${\displaystyle X}$ is a locally convex space and if ${\displaystyle {\mathcal {P}}}$ is a drove of continuous seminorms on ${\displaystyle X}$ , and so ${\displaystyle {\mathcal {P}}}$ is chosen a base of continuous seminorms if information technology is a base of seminorms for the collection of all continuous seminorms on ${\displaystyle X}$ .^[eight] Explicitly, this means that for all continuous seminorms ${\displaystyle p}$ on ${\displaystyle X}$ , in that location exists a ${\displaystyle q\in {\mathcal {P}}}$ and a real ${\displaystyle r>0}$ such that ${\displaystyle p\leq rq.}$ ^[8]

If ${\displaystyle {\mathcal {P}}}$ is a base of continuous seminorms for a locally convex TVS ${\displaystyle X}$ so the family unit of all sets of the grade ${\displaystyle \{x\in Ten:q(x)<r\}}$ as ${\displaystyle q}$ varies over ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle r}$ varies over the positive real numbers, is a base of neighborhoods of the origin in ${\displaystyle 10}$ (not just a subbasis, so in that location is no need to take finite intersections of such sets).^[eight]

A family unit ${\displaystyle {\mathcal {P}}}$ of seminorms on a vector space ${\displaystyle X}$ is called saturated if for whatever ${\displaystyle p}$ and ${\displaystyle q}$ in ${\displaystyle {\mathcal {P}},}$ the seminorm defined past ${\displaystyle x\mapsto \max\{p(x),q(x)\}}$ belongs to ${\displaystyle {\mathcal {P}}.}$

If ${\displaystyle {\mathcal {P}}}$ is a saturated family of continuous seminorms that induces the topology on ${\displaystyle X}$ and so the drove of all sets of the course ${\displaystyle \{x\in Ten:p(10)<r\}}$ equally ${\displaystyle p}$ ranges over ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle r}$ ranges over all positive real numbers, forms a neighborhood basis at the origin consisting of convex open up sets;^[eight] This forms a basis at the origin rather than merely a subbasis so that in particular, there is no need to accept finite intersections of such sets.^[8]

Footing of norms [edit]

The following theorem implies that if ${\displaystyle X}$ is a locally convex space so the topology of ${\displaystyle Ten}$ can be a defined past a family of continuous norms on ${\displaystyle 10}$ (a norm is a seminorm ${\displaystyle south}$ where ${\displaystyle s(x)=0}$ implies ${\displaystyle x=0}$ ) if and only if there exists at least one continuous norm on ${\displaystyle X}$ .^[9] This is because the sum of a norm and a seminorm is a norm and so if a locally convex space is defined by some family ${\displaystyle {\mathcal {P}}}$ of seminorms (each is which is necessarily continuous) then the family ${\displaystyle {\mathcal {P}}+n:=\{p+n:p\in {\mathcal {P}}\}}$ of (too continuous) norms obtained by calculation some given continuous norm ${\displaystyle n}$ to each element, will necessarily be a family of norms that defines this aforementioned locally convex topology. If there exists a continuous norm on a topological vector space ${\displaystyle Ten}$ and then ${\displaystyle X}$ is necessarily Hausdorff just the antipodal is non in general true (not even for locally convex spaces or Fréchet spaces).

Nets [edit]

Suppose that the topology of a locally convex space ${\displaystyle X}$ is induced by a family ${\displaystyle {\mathcal {P}}}$ of continuous seminorms on ${\displaystyle X}$ . If ${\displaystyle x\in X}$ and if ${\displaystyle x_{\bullet }=\left(x_{i}\correct)_{i\in I}}$ is a net in ${\displaystyle X}$ , and so ${\displaystyle x_{\bullet }\to ten}$ in ${\displaystyle X}$ if and merely if for all ${\displaystyle p\in {\mathcal {P}},}$ ${\displaystyle p\left(x_{\bullet }-x\right)=\left(p\left(x_{i}\right)-x\correct)_{i\in I}\to 0.}$ ^[11] Moreover, if ${\displaystyle x_{\bullet }}$ is Cauchy in ${\displaystyle X}$ , and so then is ${\displaystyle p\left(x_{\bullet }\right)=\left(p\left(x_{i}\right)\right)_{i\in I}}$ for every ${\displaystyle p\in {\mathcal {P}}.}$ ^[11]

Equivalence of definitions [edit]

Although the definition in terms of a neighborhood base gives a better geometric pic, 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 ${\displaystyle \varepsilon }$ -balls is the triangle inequality.

For an absorbing set ${\displaystyle C}$ such that if ${\displaystyle 10\in C,}$ then ${\displaystyle tx\in C}$ whenever ${\displaystyle 0\leq t\leq one,}$ define the Minkowski functional of ${\displaystyle C}$ to be

$${\displaystyle \mu _{C}(ten)=\inf\{r>0:x\in rC\}.}$$

From this definition it follows that ${\displaystyle \mu _{C}}$ is a seminorm if ${\displaystyle C}$ is balanced and convex (it is also absorbent by assumption). Conversely, given a family of seminorms, the sets

$${\displaystyle \left\{x:p_{\alpha _{1}}(x)<\varepsilon _{1},\ldots ,p_{\alpha _{due north}}(x)<\varepsilon _{n}\right\}}$$

form a base of operations of convex absorbent counterbalanced sets.

Ways of defining a locally convex topology [edit]

Theorem^[seven]  —Suppose that ${\displaystyle X}$ is a (real or complex) vector space and permit ${\displaystyle {\mathcal {L}}}$ exist a not-empty collection of convex, counterbalanced, and absorbing subsets of ${\displaystyle X}$ . Then the set up of all of all positive scalar multiples of finite intersections of sets in ${\displaystyle {\mathcal {50}}}$ forms a neighborhood base at the origin for a locally convex TVS topology on ${\displaystyle Ten}$ .

Further definitions [edit]

• A family of seminorms ${\displaystyle \left(p_{\alpha }\correct)_{\alpha }}$ is called total or separated or is said to dissever points if whenever ${\displaystyle p_{\alpha }(x)=0}$ holds for every ${\displaystyle \alpha }$ then ${\displaystyle x}$ is necessarily ${\displaystyle 0.}$ A locally convex space is Hausdorff if and but if it has a separated family of seminorms. Many authors have the Hausdorff criterion in the definition.
• A pseudometric is a generalization of a metric which does not satisfy the condition that ${\displaystyle d(x,y)=0}$ only when ${\displaystyle x=y.}$ 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 so given past

  $${\displaystyle d(x,y)=\sum _{northward}^{\infty }{\frac {i}{ii^{northward}}}{\frac {p_{due north}(x-y)}{1+p_{north}(10-y)}}}$$

  

  (where the ${\displaystyle i/2^{n}}$ tin be replaced by any positive summable sequence ${\displaystyle a_{n}}$ ). This pseudometric is translation-invariant, just not homogeneous, pregnant ${\displaystyle d(kx,ky)\neq |k|d(ten,y),}$ and therefore does not define a (pseudo)norm. The pseudometric is an honest metric if and but if the family of seminorms is separated, since this is the case if and merely if the space is Hausdorff. If furthermore the space is complete, the infinite is called a Fréchet space.
• Every bit with any topological vector space, a locally convex infinite is likewise 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 ${\displaystyle \left(x_{a}\correct)_{a\in A}}$ such that for every ${\displaystyle r>0}$ and every seminorm ${\displaystyle p_{\alpha },}$ there exists some index ${\displaystyle c\in A}$ such that for all indices ${\displaystyle a,b\geq c,}$ ${\displaystyle p_{\blastoff }\left(x_{a}-x_{b}\correct)<r.}$ In other words, the net must exist 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 past an uncountable family of pseudometrics. Sequences, which are countable by definition, cannot suffice to narrate convergence in such spaces. A locally convex infinite is complete if and only if every Cauchy cyberspace converges.
• A family of seminorms becomes a preordered fix nether the relation ${\displaystyle p_{\alpha }\leq p_{\beta }}$ if and just if there exists an ${\displaystyle G>0}$ such that for all ${\displaystyle x,}$ ${\displaystyle p_{\alpha }(x)\leq Mp_{\beta }(x).}$ One says it is a directed family of seminorms if the family is a directed prepare with addition equally the join, in other words if for every ${\displaystyle \alpha }$ and ${\displaystyle \beta ,}$ there is a ${\displaystyle \gamma }$ such that ${\displaystyle p_{\alpha }+p_{\beta }\leq p_{\gamma }.}$ Every family of seminorms has an equivalent directed family unit, significant 1 which defines the aforementioned topology. Indeed, given a family unit ${\displaystyle \left(p_{\alpha }(x)\right)_{\alpha \in I},}$ allow ${\displaystyle \Phi }$ exist the prepare of finite subsets of ${\displaystyle I}$ and then for every ${\displaystyle F\in \Phi }$ define

  $${\displaystyle q_{F}=\sum _{\alpha \in F}p_{\alpha }.}$$

  

  I may check that ${\displaystyle \left(q_{F}\right)_{F\in \Phi }}$ is an equivalent directed family.
• If the topology of the space is induced from a unmarried seminorm, then the infinite is seminormable. Whatever locally convex space with a finite family unit of seminorms is seminormable. Moreover, if the space is Hausdorff (the family is separated), then the space is normable, with norm given past the sum of the seminorms. In terms of the open sets, a locally convex topological vector infinite is seminormable if and only if the origin has a bounded neighborhood.

Sufficient weather [edit]

Hahn–Banach extension property [edit]

Let ${\displaystyle Ten}$ be a TVS. Say that a vector subspace ${\displaystyle M}$ of ${\displaystyle X}$ has the extension property if any continuous linear functional on ${\displaystyle M}$ tin can be extended to a continuous linear functional on ${\displaystyle X}$ .^[12] Say that ${\displaystyle X}$ has the Hahn-Banach extension holding (HBEP) if every vector subspace of ${\displaystyle X}$ has the extension property.^[12]

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

Theorem^[12] (Kalton)  —Every complete metrizable TVS with the Hahn-Banach extension holding is locally convex.

If a vector space ${\displaystyle 10}$ has uncountable dimension and if we endow information technology with the finest vector topology then this is a TVS with the HBEP that is neither locally convex or metrizable.^[12]

Properties [edit]

Throughout, ${\displaystyle {\mathcal {P}}}$ is a family of continuous seminorms that generate the topology of ${\displaystyle 10}$ .

Topological properties [edit]

Topological properties of convex subsets [edit]

• The interior and closure of a convex subset of a TVS is once more convex.^[17]
• The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex ready is over again convex.^[17]
• If ${\displaystyle C}$ is a convex set with non-empty interior, then the closure of ${\displaystyle C}$ is equal to the closure of the interior of ${\displaystyle C}$ ; furthermore, the interior of ${\displaystyle C}$ is equal to the interior of the closure of ${\displaystyle C.}$ ^{[17] [18]}
• If ${\displaystyle C}$ is a convex subset of a TVS ${\displaystyle X}$ (non necessarily Hausdorff), ${\displaystyle 10}$ belongs to the interior of ${\displaystyle S}$ , and ${\displaystyle y}$ belongs to the closure of ${\displaystyle South}$ , then the open line segment articulation ${\displaystyle ten}$ and ${\displaystyle y}$ (that is, ${\displaystyle \{tx+(ane-t)y:0<t<1\}}$ ) belongs to the interior of ${\displaystyle S}$ .^{[18] [19]}
• If ${\displaystyle Ten}$ is a locally convex space (not necessarily Hausdorff), ${\displaystyle M}$ is a closed vector subspace of ${\displaystyle X}$ , V is a convex neighborhood of the origin in ${\displaystyle G}$ , and if ${\displaystyle z\in X}$ is a vector not in ${\displaystyle V,}$ then in that location exists a convex neighborhood U of 0 in ${\displaystyle X}$ such that ${\displaystyle V=U\cap G}$ and ${\displaystyle z\non \in U.}$ ^[17]
• The closure of a convex subset of a Hausdorff locally convex TVS ${\displaystyle X}$ is the same for all locally convex Hausdorff TVS topologies on ${\displaystyle X}$ that are compatible with duality between ${\displaystyle X}$ and its continuous dual infinite.^[20]
• In a locally convex infinite, 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]
• In a locally convex space, convex hulls of divisional sets are bounded. This is not true for TVSs in general.^[21]
• In a Fréchet space, the airtight convex hull of a compact set is compact.^[22]
• In a locally convex space, any linear combination of totally divisional sets is totally bounded.^[21]

Properties of convex hulls [edit]

For whatever subset ${\displaystyle S}$ of a TVS ${\displaystyle X}$ , the convex hull (resp. closed convex hull, balanced hull, resp. convex balanced hull) of ${\displaystyle S}$ , denoted by ${\displaystyle \operatorname {co} Southward}$ (resp. ${\displaystyle {\overline {\operatorname {co} }}S,}$ ${\displaystyle \operatorname {bal} S,}$ ${\displaystyle \operatorname {cobal} S}$ ), is the smallest convex (resp. closed convex, balanced, convex balanced) subset of ${\displaystyle X}$ containing ${\displaystyle South.}$

• In a quasi-complete locally convex TVS, the closure of the convex hull of a compact subset is once again compact.
• In a Hausdorff locally convex TVS, the convex hull of a precompact set is again precompact.^[23] Consequently, in a complete locally convex Hausdorff TVS, the airtight convex hull of a compact subset is again compact.^[24]
• In whatsoever TVS, the convex hull of a finite marriage of compact convex sets is compact (and convex).^[vii]
  • This implies that in whatever Hausdorff TVS, the convex hull of a finite union of compact convex sets is closed (in addition to being compact and convex); in detail, the convex hull of such a union is equal to the closed convex hull of that wedlock.
  • In full general, the closed convex hull of a meaty set up is not necessarily compact.
  • In any non-Hausdorff TVS, there be subsets that are meaty (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 airtight convex counterbalanced hull of that set.^[25]
• The balanced hull of a convex set is not necessarily convex.
• If ${\displaystyle C}$ and ${\displaystyle D}$ are convex subsets of a topological vector space ${\displaystyle 10}$ and if ${\displaystyle \operatorname {co} (C\cup D),}$ and then there exist ${\displaystyle c\in C,}$ ${\displaystyle d\in D,}$ and a real number ${\displaystyle r}$ satisfying ${\displaystyle 0\leq r\leq one}$ such that ${\displaystyle x=rc+(1-r)d.}$ ^[17]
• If ${\displaystyle M}$ is a vector subspace of a TVS ${\displaystyle 10}$ , ${\displaystyle C}$ a convex subset of ${\displaystyle M}$ , and ${\displaystyle D}$ a convex subset of ${\displaystyle X}$ such that ${\displaystyle D\cap M\subseteq C,}$ and so ${\displaystyle C=One thousand\cap \operatorname {co} (C\cup D).}$ ^[17]
• Call back that the smallest balanced subset of ${\displaystyle X}$ containing a set ${\displaystyle S}$ is called the counterbalanced hull of ${\displaystyle S}$ and is denoted by ${\displaystyle \operatorname {bal} South.}$ For any subset ${\displaystyle S}$ of ${\displaystyle X}$ , the convex balanced hull of ${\displaystyle Due south}$ , denoted by ${\displaystyle \operatorname {cobal} South,}$ is the smallest subset of ${\displaystyle X}$ containing ${\displaystyle S}$ that is convex and balanced.^[26] The convex balanced hull of ${\displaystyle S}$ is equal to the convex hull of the balanced hull of ${\displaystyle Southward}$ (i.e. ${\displaystyle \operatorname {cobal} S=\operatorname {co} (\operatorname {bal} South)}$ ), simply the convex balanced hull of ${\displaystyle S}$ is not necessarily equal to the balanced hull of the convex hull of ${\displaystyle S}$ (that is, ${\displaystyle \operatorname {cobal} S}$ is not necessarily equal to ${\displaystyle \operatorname {bal} (\operatorname {co} Due south)}$ ).^[26]
• If ${\displaystyle A,B\subseteq X}$ are subsets of a TVS ${\displaystyle X}$ and if ${\displaystyle r}$ is a scalar then ${\displaystyle \operatorname {co} (A\cup B)=\operatorname {co} (A)\cup \operatorname {co} (B),}$ ${\displaystyle \operatorname {co} (rA)=r\operatorname {co} A,}$ and ${\displaystyle {\overline {\operatorname {co} }}(rA)=r{\overline {\operatorname {co} }}(A).}$ Moreover, if ${\displaystyle {\overline {\operatorname {co} }}(A)}$ is compact so ${\displaystyle {\overline {\operatorname {co} }}(A+B)={\overline {\operatorname {co} }}(A)+{\overline {\operatorname {co} }}(B)}$ ^[27]
• If ${\displaystyle A,B\subseteq 10}$ are subsets of a TVS ${\displaystyle X}$ whose airtight convex hulls are meaty, then ${\displaystyle {\overline {\operatorname {co} }}(A\cup B)={\overline {\operatorname {co} }}\left({\overline {\operatorname {co} }}(A)\cup {\overline {\operatorname {co} }}(B)\right).}$ ^[27]
• If ${\displaystyle South}$ is a convex prepare in a complex vector infinite ${\displaystyle X}$ and at that place exists some ${\displaystyle z\in X}$ such that ${\displaystyle z,iz,-z,-iz\in S,}$ and so ${\displaystyle rz+siz\in Due south}$ for all real ${\displaystyle r,s}$ such that ${\displaystyle |r|+|s|\leq 1.}$ In particular, ${\displaystyle az\in S}$ for all scalars ${\displaystyle a}$ such that ${\displaystyle |a|^{2}\leq {\frac {1}{two}}.}$

Examples and nonexamples [edit]

Finest and coarsest locally convex topology [edit]

Coarsest vector topology [edit]

Any vector space ${\displaystyle X}$ endowed with the trivial topology (likewise called the indiscrete topology) is a locally convex TVS (and of grade, it is the coarsest such topology). This topology is Hausdorff if and only ${\displaystyle X=\{0\}.}$ The indiscrete topology makes any vector space into a consummate pseudometrizable locally convex TVS.

In contrast, the discrete topology forms a vector topology on ${\displaystyle X}$ if and just ${\displaystyle X=\{0\}.}$ This follows from the fact that every topological vector space is a connected space.

Finest locally convex topology [edit]

If ${\displaystyle Ten}$ is a real or circuitous vector space and if ${\displaystyle {\mathcal {P}}}$ is the set of all seminorms on ${\displaystyle X}$ and so the locally convex TVS topology, denoted by ${\displaystyle \tau _{\operatorname {lc} },}$ that ${\displaystyle {\mathcal {P}}}$ induces on ${\displaystyle X}$ is called the finest locally convex topology on ${\displaystyle X.}$ ^[28] This topology may also be described every bit the TVS-topology on ${\displaystyle X}$ having every bit a neighborhood base of operations at the origin the set of all arresting disks in ${\displaystyle X.}$ ^[28] Whatsoever locally convex TVS-topology on ${\displaystyle X}$ is necessarily a subset of ${\displaystyle \tau _{\operatorname {lc} }.}$ ${\displaystyle \left(X,\tau _{\operatorname {lc} }\right)}$ is Hausdorff.^[fifteen] Every linear map from ${\displaystyle \left(Ten,\tau _{\operatorname {lc} }\right)}$ into another locally convex TVS is necessarily continuous.^[xv] In item, every linear functional on ${\displaystyle \left(X,\tau _{\operatorname {lc} }\right)}$ is continuous and every vector subspace of ${\displaystyle 10}$ is closed in ${\displaystyle \left(X,\tau _{\operatorname {lc} }\right)}$ ;^[15] therefore, if ${\displaystyle Ten}$ is space dimensional and then ${\displaystyle \left(X,\tau _{\operatorname {lc} }\right)}$ is not pseudometrizable (and thus not metrizable).^[28] Moreover, ${\displaystyle \tau _{\operatorname {lc} }}$ is the only Hausdorff locally convex topology on ${\displaystyle X}$ with the property that any linear map from information technology into any Hausdorff locally convex infinite is continuous.^[29] The space ${\displaystyle \left(X,\tau _{\operatorname {lc} }\right)}$ is a bornological infinite.^[30]

Examples of locally convex spaces [edit]

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 exist taken to be the single norm. Every Banach space is a consummate Hausdorff locally convex space, in particular, the ${\displaystyle 50^{p}}$ spaces with ${\displaystyle p\geq ane}$ are locally convex.

More than by and large, every Fréchet space is locally convex. A Fréchet space can be defined equally a complete locally convex space with a separated countable family of seminorms.

The infinite ${\displaystyle \mathbb {R} ^{\omega }}$ of real valued sequences with the family of seminorms given by

$${\displaystyle p_{i}\left(\left\{x_{north}\right\}_{northward}\correct)=\left|x_{i}\right|,\qquad i\in \mathbb {N} }$$

is locally convex. The countable family of seminorms is complete and separable, so this is a Fréchet space, which is non normable. This is also the limit topology of the spaces ${\displaystyle \mathbb {R} ^{n},}$ embedded in ${\displaystyle \mathbb {R} ^{\omega }}$ in the natural way, by completing finite sequences with infinitely many ${\displaystyle 0.}$

Given any vector space ${\displaystyle X}$ and a collection ${\displaystyle F}$ of linear functionals on it, ${\displaystyle X}$ can be fabricated into a locally convex topological vector infinite by giving information technology the weakest topology making all linear functionals in ${\displaystyle F}$ continuous. This is known equally the weak topology or the initial topology determined by ${\displaystyle F.}$ The drove ${\displaystyle F}$ may exist the algebraic dual of ${\displaystyle Ten}$ or any other collection. The family unit of seminorms in this case is given by ${\displaystyle p_{f}(x)=|f(ten)|}$ for all ${\displaystyle f}$ in ${\displaystyle F.}$

Spaces of differentiable functions give other non-normable examples. Consider the infinite of smooth functions ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {C} }$ such that ${\displaystyle \sup _{x}\left|ten^{a}D_{b}f\correct|<\infty ,}$ where ${\displaystyle a}$ and ${\displaystyle B}$ are multiindices. The family unit of seminorms divers by ${\displaystyle p_{a,b}(f)=\sup _{ten}\left|x^{a}D_{b}f(x)\right|}$ is separated, and countable, and the infinite is complete, so this metrizable space is a Fréchet infinite. It is known as the Schwartz space, or the space of functions of rapid decrease, and its dual space is the infinite of tempered distributions.

An important function space in functional analysis is the space ${\displaystyle D(U)}$ of smoothen functions with compact support in ${\displaystyle U\subseteq \mathbb {R} ^{n}.}$ A more than detailed construction is needed for the topology of this infinite considering the space ${\displaystyle C_{0}^{\infty }(U)}$ is not complete in the uniform norm. The topology on ${\displaystyle D(U)}$ is defined as follows: for whatever stock-still compact fix ${\displaystyle K\subseteq U,}$ the space ${\displaystyle C_{0}^{\infty }(K)}$ of functions ${\displaystyle f\in C_{0}^{\infty }}$ with ${\displaystyle \operatorname {supp} (f)\subseteq G}$ is a Fréchet space with countable family of seminorms ${\displaystyle \|f\|_{one thousand}=\sup _{k\leq m}\sup _{x}\left|D^{one thousand}f(10)\right|}$ (these are actually norms, and the completion of the space ${\displaystyle C_{0}^{\infty }(Thou)}$ with the ${\displaystyle \|\cdot \|_{g}}$ norm is a Banach space ${\displaystyle D^{thousand}(K)}$ ). Given any collection ${\displaystyle \left(K_{a}\right)_{a\in A}}$ of compact sets, directed past inclusion and such that their union equal ${\displaystyle U,}$ the ${\displaystyle C_{0}^{\infty }\left(K_{a}\right)}$ class a direct system, and ${\displaystyle D(U)}$ is defined to be the limit of this system. Such a limit of Fréchet spaces is known as an LF space. More than concretely, ${\displaystyle D(U)}$ is the union of all the ${\displaystyle C_{0}^{\infty }\left(K_{a}\right)}$ with the strongest locally convex topology which makes each inclusion map ${\displaystyle C_{0}^{\infty }\left(K_{a}\right)\hookrightarrow D(U)}$ continuous. This space is locally convex and complete. However, it is not metrizable, and so it is non a Fréchet infinite. The dual space of ${\displaystyle D\left(\mathbb {R} ^{n}\right)}$ is the space of distributions on ${\displaystyle \mathbb {R} ^{n}.}$

More abstractly, given a topological infinite ${\displaystyle 10,}$ the infinite ${\displaystyle C(X)}$ of continuous (non necessarily bounded) functions on ${\displaystyle 10}$ can be given the topology of uniform convergence on compact sets. This topology is defined past semi-norms ${\displaystyle \varphi _{K}(f)=\max\{|f(10)|:x\in K\}}$ (equally ${\displaystyle K}$ varies over the directed set of all meaty subsets of ${\displaystyle X}$ ). When ${\displaystyle X}$ is locally compact (for case, an open prepare in ${\displaystyle \mathbb {R} ^{n}}$ ) the Stone–Weierstrass theorem applies—in the case of existent-valued functions, whatsoever subalgebra of ${\displaystyle C(Ten)}$ that separates points and contains the constant functions (for instance, the subalgebra of polynomials) is dense.

Examples of spaces defective local convexity [edit]

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

• The spaces ${\displaystyle L^{p}([0,1])}$ for ${\displaystyle 0<p<1}$ are equipped with the F-norm

  $${\displaystyle \|f\|_{p}^{p}=\int _{0}^{1}|f(x)|^{p}\,dx.}$$

  

  They are not locally convex, since the just convex neighborhood of zero is the whole space. More mostly the spaces ${\displaystyle 50^{p}(\mu )}$ with an atomless, finite measure ${\displaystyle \mu }$ and ${\displaystyle 0<p<one}$ are not locally convex.
• The space of measurable functions on the unit interval ${\displaystyle [0,i]}$ (where we identify two functions that are equal almost everywhere) has a vector-infinite topology divers by the translation-invariant metric (which induces the convergence in measure of measurable functions; for random variables, convergence in measure out is convergence in probability):

  $${\displaystyle d(f,g)=\int _{0}^{one}{\frac {|f(10)-m(x)|}{1+|f(10)-g(x)|}}\,dx.}$$

  

  This infinite is ofttimes denoted ${\displaystyle L_{0}.}$

Both examples accept the property that any continuous linear map to the real numbers is ${\displaystyle 0.}$ In item, their dual infinite is niggling, that is, information technology contains only the nothing functional.

Continuous mappings [edit]

Theorem^[31]  —Permit ${\displaystyle T:X\to Y}$ be a linear operator between TVSs where ${\displaystyle Y}$ is locally convex (note that ${\displaystyle X}$ need not be locally convex). And so ${\displaystyle T}$ is continuous if and simply if for every continuous seminorm ${\displaystyle q}$ on ${\displaystyle Y}$ , there exists a continuous seminorm ${\displaystyle p}$ on ${\displaystyle X}$ such that ${\displaystyle q\circ T\leq p.}$

Because locally convex spaces are topological spaces too every bit vector spaces, the natural functions to consider between 2 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 than familiar boundedness condition found for Banach spaces.

Given locally convex spaces ${\displaystyle X}$ and ${\displaystyle Y}$ with families of seminorms ${\displaystyle \left(p_{\alpha }\right)_{\alpha }}$ and ${\displaystyle \left(q_{\beta }\right)_{\beta }}$ respectively, a linear map ${\displaystyle T:X\to Y}$ is continuous if and only if for every ${\displaystyle \beta ,}$ in that location be ${\displaystyle \alpha _{ane},\ldots ,\blastoff _{n}}$ and ${\displaystyle M>0}$ such that for all ${\displaystyle v\in X,}$

$${\displaystyle q_{\beta }(Television)\leq M\left(p_{\alpha _{1}}(5)+\dotsb +p_{\alpha _{n}}(five)\right).}$$

In other words, each seminorm of the range of ${\displaystyle T}$ is divisional in a higher place by some finite sum of seminorms in the domain. If the family ${\displaystyle \left(p_{\alpha }\right)_{\alpha }}$ is a directed family, and it can always exist chosen to be directed as explained above, then the formula becomes fifty-fifty simpler and more familiar:

$${\displaystyle q_{\beta }(Tv)\leq Mp_{\blastoff }(five).}$$

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

Linear functionals [edit]

Theorem^[31]  —If ${\displaystyle X}$ is a TVS (not necessarily locally convex) and if ${\displaystyle f}$ is a linear functional on ${\displaystyle X}$ , then ${\displaystyle f}$ is continuous if and just if there exists a continuous seminorm ${\displaystyle p}$ on ${\displaystyle Ten}$ such that ${\displaystyle |f|\leq p.}$

If ${\displaystyle X}$ is a real or complex vector infinite, ${\displaystyle f}$ is a linear functional on ${\displaystyle X}$ , and ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$ , then ${\displaystyle |f|\leq p}$ if and but if ${\displaystyle f\leq p.}$ ^[32] If ${\displaystyle f}$ is a non-0 linear functional on a real vector infinite ${\displaystyle X}$ and if ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$ , then ${\displaystyle f\leq p}$ if and but if ${\displaystyle f^{-1}(1)\cap \{x\in Ten:p(x)<1\}=\varnothing .}$ ^[15]

Multilinear maps [edit]

Permit ${\displaystyle n\geq i}$ be an integer, ${\displaystyle X_{1},\ldots ,X_{n}}$ be TVSs (not necessarily locally convex), let ${\displaystyle Y}$ exist a locally convex TVS whose topology is determined by a family ${\displaystyle {\mathcal {Q}}}$ of continuous seminorms, and let ${\displaystyle M:\prod _{i=ane}^{n}X_{i}\to Y}$ be a multilinear operator that is linear in each of its ${\displaystyle n}$ coordinates. The post-obit are equivalent:

1. ${\displaystyle M}$ is continuous.
2. For every ${\displaystyle q\in {\mathcal {Q}},}$ in that location be continuous seminorms ${\displaystyle p_{1},\ldots ,p_{northward}}$ on ${\displaystyle X_{1},\ldots ,X_{n},}$ respectively, such that ${\displaystyle q(M(x))\leq p_{1}\left(x_{1}\right)\cdots p_{northward}\left(x_{n}\right)}$ for all ${\displaystyle ten=\left(x_{1},\ldots ,x_{n}\right)\in \prod _{i=1}^{northward}X_{i}.}$ ^[fifteen]
3. For every ${\displaystyle q\in {\mathcal {Q}},}$ there exists some neighborhood of the origin in ${\displaystyle \prod _{i=ane}^{northward}X_{i}}$ on which ${\displaystyle q\circ Grand}$ is bounded.^[15]

Run across also [edit]

• Convex set – In geometry, set that intersects every line into a single line segment
• Krein–Milman theorem – On when a space equals the airtight 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
• Seminorm
• Sublinear functional
• Topological group – Group that is a topological space with continuous group activeness
• Topological vector space – Vector infinite with a notion of nearness
• Vector space – Algebraic structure in linear algebra

Notes [edit]

1. ^ Hausdorff, F. Grundzüge der Mengenlehre (1914)
2. ^ von Neumann, J. Collected works. Vol 2. pp. 94–104
3. ^ Dieudonne, J. History of Functional Assay Chapter Viii. Section 1.
4. ^ von Neumann, J. Collected works. Vol Two. pp. 508–527
5. ^ Dieudonne, J. History of Functional Analysis Chapter Eight. Section 2.
6. ^ Banach, S. Theory of linear operations p. 75. Ch. Viii. Sec. 3. Theorem 4., translated from Theorie des operations lineaires (1932)
7. ^ ^{a b c d e f g h i} Narici & Beckenstein 2011, pp. 67–113.
8. ^ ^{a b c d due east f} Narici & Beckenstein 2011, p. 122.
9. ^ Jarchow 1981, p. 130.
10. ^ Jarchow 1981, pp. 129–130.
11. ^ ^{a b} Narici & Beckenstein 2011, p. 126.
12. ^ ^{a b c d} Narici & Beckenstein 2011, pp. 225–273.
13. ^ Narici & Beckenstein 2011, pp. 177–220.
14. ^ Narici & Beckenstein 2011, p. 149.
15. ^ ^{a b c d e f g} Narici & Beckenstein 2011, pp. 149–153.
16. ^ Narici & Beckenstein 2011, pp. 115–154.
17. ^ ^{a b c d e f} Trèves 2006, p. 126.
18. ^ ^{a b} Schaefer & Wolff 1999, p. 38.
19. ^ Conway 1990, p. 102.
20. ^ Trèves 2006, p. 370.
21. ^ ^{a b} Narici & Beckenstein 2011, pp. 155–176.
22. ^ Rudin 1991, p. 7.
23. ^ Trèves 2006, p. 67.
24. ^ Trèves 2006, p. 145.
25. ^ Trèves 2006, p. 362.
26. ^ ^{a b} Trèves 2006, p. 68.
27. ^ ^{a b} Dunford 1988, p. 415.
28. ^ ^{a b c} Narici & Beckenstein 2011, pp. 125–126.
29. ^ Narici & Beckenstein 2011, p. 476.
30. ^ Narici & Beckenstein 2011, p. 446.
31. ^ ^{a b} Narici & Beckenstein 2011, pp. 126–128.
32. ^ Narici & Beckenstein 2011, pp. 126-–128.

References [edit]

• Berberian, Sterling M. (1974). Lectures in Functional Assay and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN978-0-387-90081-0. OCLC 878109401.
• Bourbaki, Nicolas (1987) [1981]. Sur certains espaces vectoriels topologiques [Topological Vector Spaces: Chapters 1–5]. Annales de l'Institut Fourier. Éléments de mathématique. Vol. 2. Translated by Eggleston, H.G.; Madan, Due south. Berlin New York: Springer-Verlag. ISBN978-3-540-42338-6. OCLC 17499190.
• Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN978-0-387-97245-9. OCLC 21195908.
• Dunford, Nelson (1988). Linear operators (in Romanian). New York: Interscience Publishers. ISBN0-471-60848-3. OCLC 18412261.
• Edwards, Robert East. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN978-0-486-68143-6. OCLC 30593138.
• Grothendieck, Alexander (1973). Topological Vector Spaces . Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN978-0-677-30020-7. OCLC 886098.
• Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN978-3-519-02224-4. 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 & Concern Media. ISBN978-3-642-64988-2. MR 0248498. OCLC 840293704.
• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and practical mathematics (2d ed.). Boca Raton, FL: CRC Press. ISBN978-1584888666. OCLC 144216834.
• Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN978-0-521-29882-7. OCLC 589250.
• Rudin, Walter (1991). Functional Analysis. International Serial in Pure and Applied Mathematics. Vol. viii (2d ed.). New York, NY: McGraw-Colina Science/Technology/Math. ISBN978-0-07-054236-5. 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. ISBN978-i-4612-7155-0. OCLC 840278135.
• Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN978-0-8247-8643-4. OCLC 24909067.
• Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN978-0-486-45352-1. OCLC 853623322.
• Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN978-0-486-49353-iv. OCLC 849801114.

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Source: https://en.wikipedia.org/wiki/Locally_convex_topological_vector_space

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