Locally convex topological vector space explained
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.
History
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).
Definition
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.
Definition via convex sets
A topological vector space (TVS) is called if it has a neighborhood basis (that is, a local base) at the origin consisting of balanced, convex sets. The term is sometimes shortened to or .
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". Note that in the TVS
,
belongs to
ball centered at the origin of radius
, but
does not belong; indeed,
C is a
cone, but balanced.
- 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.
- Absolutely convex or a 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 sets (that is, disks), where this neighborhood basis can further be chosen to also consist entirely of open sets or entirely of closed sets. 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 convex. It is possible for a TVS to have 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.
Definition via seminorms
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
Definition (second version): A locally convex space is defined to be a vector space
along with a
family
of seminorms on
Seminorm topology
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 be normable (that is, to have its topology be induced by a single norm).
Basis and subbases
An open set in
has the form
, where
is a positive real number. The family of preimages
p-1\left([0,r)\right)=\{x\inX:p(x)<r\}
as
ranges over a family of seminorms
and
ranges over the positive real numbersis 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 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
Bases of seminorms and saturated families
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 continuous seminorms on
. Explicitly, this means that for all continuous seminorms
on
, there exists a
and a real
such that
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 of neighborhoods of the origin in
(not just a subbasis, so there is no need to take finite intersections of such sets).
[7] 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; This forms a basis at the origin rather than merely a subbasis so that in particular, there is need to take finite intersections of such sets.
=Basis of norms
=
The following theorem implies that if
is a locally convex space then the topology of
can be a defined by a family of continuous on
(a
norm is a
seminorm
where
implies
) if and only if there exists continuous on
. 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).
Nets
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
p\left(x\bull-x\right)=\left(p\left(xi\right)-x\right)i\to0.
Moreover, if
is Cauchy in
, then so is
p\left(x\bull\right)=\left(p\left(xi\right)\right)i
for every
Equivalence of definitions
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.
Ways of defining a locally convex topology
Example: auxiliary normed spaces
If
is
convex and
absorbing in
then the
symmetric set
will be convex and
balanced (also known as an or a) 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
\left\{\tfrac{1}{2},\tfrac{1}{3},\tfrac{1}{4},\ldots\right\}
(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 .
Further definitions
\left(p\alpha\right)\alpha
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.
such that for every
and every seminorm
there exists some index
such that for all indices
p\alpha\left(xa-xb\right)<r.
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
p\alpha(x)\leqMp\beta(x).
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
p\alpha+p\beta\leqp\gamma.
Every family of seminorms has an equivalent directed family, meaning one which defines the same topology. Indeed, given a family
\left(p\alpha(x)\right)\alpha,
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.
Sufficient conditions
Hahn–Banach extension property
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
. Say that
has the
Hahn-Banach extension property (
HBEP) if every vector subspace of
has the extension property.
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.
Properties
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
The closure of
in
is equal to
} p^(0).
Topology of Hausdorff locally convex spaces
Every Hausdorff locally convex space is homeomorphic to a vector subspace of a product of Banach spaces. 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).
Properties of convex subsets
Algebraic properties of convex subsets
A subset
is convex if and only if
for all
or equivalently, if and only if
for all positive real
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
(sC)\cap(tC)=(min\{s,t\})C.
The Minkowski sum of two convex sets is convex; furthermore, the scalar multiple of a convex set is again convex.
Topological properties of convex subsets
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
z+\{y\inY:p(y)<1\}=\{y\inY:p(y-z)<1\}
for some
and some positive continuous
sublinear functional
on
- The interior and closure of a convex subset of a TVS is again convex.
- 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
- 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.
is convex and
then
t\operatorname{Int}C+(1-t)\operatorname{cl}C~\subseteq~\operatorname{Int}C.
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,
\{tx+(1-t)y:0<t<1\}\subseteq\operatorname{int}XC.
[8]
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 in
then there exists a convex neighborhood
of the origin in
such that
and
- The closure of a convex subset of a locally convex Hausdorff space
is the same for locally convex Hausdorff TVS topologies on
that are compatible with
duality between
and its continuous dual space.
- In a locally convex space, the convex hull and the disked hull of a totally bounded set is totally bounded.
- In a complete locally convex space, the convex hull and the disked hull of a compact set are both compact.
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.
- In a Fréchet space, the closed convex hull of a compact set is compact.
- In a locally convex space, any linear combination of totally bounded sets is totally bounded.
Properties of convex hulls
For any subset
of a TVS
the
convex hull (respectively,
closed convex hull,
balanced hull,
convex balanced hull) of
denoted by
(respectively,
\overline{\operatorname{co}}S,
), is the smallest convex (respectively, closed convex, balanced, convex balanced) subset of
containing
- The convex hull of compact subset of a Hilbert space is necessarily closed and so also necessarily compact. For example, let
be the separable Hilbert space
of square-summable sequences with the usual norm
and let
en=(0,\ldots,0,1,0,\ldots)
be the standard
orthonormal basis (that is
at the
-coordinate). The closed set
S=\{0\}\cup\left\{\tfrac{1}{1}en,\tfrac{1}{2}e2,\tfrac{1}{3}e3,\ldots\right\}
is compact but its convex hull
is a closed set because
h:=
\tfrac{1}{2n}\tfrac{1}{n}en
belongs to the closure of
in
but
h\not\in\operatorname{co}S
(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
). However, like in all
complete Hausdorff locally convex spaces, the convex hull
K:=\overline{\operatorname{co}}S
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 to be compact (although it will be
precompact/totally bounded).
- In a Hausdorff locally convex space
the closed convex hull
\overline{\operatorname{co}}XS=\operatorname{cl}X\operatorname{co}S
of compact subset
is not necessarily compact although it is a precompact (also called "totally bounded") subset, which means that its closure,
of
will be compact (here
so that
if and only if
is complete); that is to say,
\operatorname{cl}\widehat{X
} \overline^X S will be compact. So for example, the closed convex hull
C:=\overline{\operatorname{co}}XS
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. Consequently, in a complete Hausdorff locally convex space, the closed convex hull of a compact subset is again compact.
- In any TVS, the convex hull of a finite union of compact convex sets is compact (and convex).
- This implies that in any Hausdorff TVS, the convex hull of a finite union of compact convex sets is (in addition to being compact and convex); in particular, the convex hull of such a union is equal to the 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.
- In any non-Hausdorff TVS, there exist subsets that are compact (and thus complete) but 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.
- The balanced hull of a convex set is necessarily convex.
- If
and
are convex subsets of a
topological vector space
and if
x\in\operatorname{co}(C\cupD),
then there exist
and a real number
satisfying
such that
is a vector subspace of a TVS
a convex subset of
and
a convex subset of
such that
then
C=M\cap\operatorname{co}(C\cupD).
- 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. The convex balanced hull of
is equal to the convex hull of the balanced hull of
(i.e.
\operatorname{cobal}S=\operatorname{co}(\operatorname{bal}S)
), but the convex balanced hull of
is necessarily equal to the balanced hull of the convex hull of
(that is,
is not necessarily equal to
\operatorname{bal}(\operatorname{co}S)
).
are subsets of a TVS
and if
is a scalar then
\operatorname{co}(A+B)=\operatorname{co}(A)+\operatorname{co}(B),
\operatorname{co}(sA)=s\operatorname{co}A,
\operatorname{co}(A\cupB)=\operatorname{co}(A)\cup\operatorname{co}(B),
and
\overline{\operatorname{co}}(sA)=s\overline{\operatorname{co}}(A).
Moreover, if
\overline{\operatorname{co}}(A)
is compact then
\overline{\operatorname{co}}(A+B)=\overline{\operatorname{co}}(A)+\overline{\operatorname{co}}(B).
However, the convex hull of a closed set need not be closed; for example, the set
\left\{(x,\pm\tanx):|x|<\tfrac{\pi}{2}\right\}
is closed in
but its convex hull is the open set
\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right) x \R.
are subsets of a TVS
whose closed convex hulls are compact, then
\overline{\operatorname{co}}(A\cupB)=\overline{\operatorname{co}}\left(\overline{\operatorname{co}}(A)\cup\overline{\operatorname{co}}(B)\right).
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
is subset of
(where
) then for every
there exist a finite subset
containing at most
points whose convex hull contains
(that is,
and
).
Examples and nonexamples
Finest and coarsest locally convex topology
Coarsest vector topology
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.
Finest locally convex topology
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
on
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
Any locally convex TVS-topology on
is necessarily a subset of
}.
\left(X,\tau\operatorname{lc
}\right) is
Hausdorff. Every linear map from
\left(X,\tau\operatorname{lc
}\right) into another locally convex TVS is necessarily continuous.In particular, every linear functional on
\left(X,\tau\operatorname{lc
}\right) is continuous and every vector subspace of
is closed in
\left(X,\tau\operatorname{lc
}\right); therefore, if
is infinite dimensional then
\left(X,\tau\operatorname{lc
}\right) is not pseudometrizable (and thus not metrizable). Moreover,
} is the Hausdorff locally convex topology on
with the property that any linear map from it into any Hausdorff locally convex space is continuous.The space
\left(X,\tau\operatorname{lc
}\right) is a
bornological space.
Examples of locally convex spaces
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
such that
\supx\left|xaDbf\right|<infty,
where
and
are
multiindices. The family of seminorms defined by
pa,b(f)=\supx\left|xaDbf(x)\right|
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
\operatorname{supp}(f)\subseteqK
is a
Fréchet space with countable family of seminorms
\|f\|m=\supk\supx\left|Dkf(x)\right|
(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 topology which makes each
inclusion map
\left(Ka\right)\hookrightarrowD(U)
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
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
\varphiK(f)=max\{|f(x)|:x\inK\}
(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.
Examples of spaces lacking local convexity
Many topological vector spaces are locally convex. Examples of spaces that lack local convexity include the following:
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.
(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.
is not locally convex.
Continuous mappings
See main article: Continuous linear map.
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
\left(p\alpha\right)\alpha
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
\left(p\alpha\right)\alpha
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.
Linear functionals
If
is a real or complex vector space,
is a linear functional on
, and
is a seminorm on
, then
if and only if
If
is a non-0 linear functional on a real vector space
and if
is a seminorm on
, then
if and only if
f-1(1)\cap\{x\inX:p(x)<1\}=\varnothing.
Multilinear maps
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
q(M(x))\leqp1\left(x1\right) … pn\left(xn\right)
for all
x=\left(x1,\ldots,xn\right)\in
Xi.
- For every
there exists some neighborhood of the origin in
on which
is bounded.
References
- .
- Book: Dunford, Nelson. Linear operators. Interscience Publishers. New York. 1988. 0-471-60848-3. 18412261. ro.
Notes and References
- 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)
- Let
be the open unit ball associated with the seminorm
and note that if
is real then
rVp=\{rx\inX:p(x)<1\}=\{z\inX:p(z)<r\}=\left\{x\inX:\tfrac{1}{r}p(x)<1\right\}=V(1/r)
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 p:=max\left\{r1p1,\ldots,rnpn\right\},
which is a continuous seminorm and moreover,
Pick
and
such that
where this inequality holds if and only if
Thus \tfrac{1}{r}Vq=Vr\subseteqVp=
\cap … \cap
,
as desired.
- Fix
so it remains to show that
w0~\stackrel{\scriptscriptstyledef
}~ r x + (1 - r) y 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 s~\stackrel{\scriptscriptstyledef
}~ \tfrac < 0 so that
Since scalar multiplication by
is a linear homeomorphism
\operatorname{cl}X\left(\tfrac{1}{s}C\right)=\tfrac{1}{s}\operatorname{cl}XC.
Since
and
it follows that x=\tfrac{1}{s}y\in\operatorname{cl}\left(\tfrac{1}{s}C\right)\cap\operatorname{int}C
where because
is open, there exists some c0\in\left(\tfrac{1}{s}C\right)\cap\operatorname{int}C,
which satisfies
Define
by x\mapstorx+(1-r)sc0=rx-rc0,
which is a homeomorphism because
The set h\left(\operatorname{int}C\right)
is thus an open subset of
that moreover contains If
then since
is convex,
and
which proves that h\left(\operatorname{int}C\right)\subseteqC.
Thus h\left(\operatorname{int}C\right)
is an open subset of
that contains the origin and is contained in
Q.E.D.