In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
and related areas of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a continuous linear operator or continuous linear mapping is a
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
linear transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
between
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s.
An operator between two
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
if and only if it is a continuous linear operator.
Continuous linear operators
Characterizations of continuity
Suppose that
is a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
between two
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs).
The following are equivalent:
- is continuous.
- is continuous at some point
- is continuous at the origin in
if
is
locally convex
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 ve ...
then this list may be extended to include:
- for every continuous
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
on there exists a continuous seminorm on such that
if
and
are both
Hausdorff locally convex spaces then this list may be extended to include:
- is weakly continuous and its
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
maps equicontinuous
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein.
In particular, the concept applies to countable fa ...
subsets of to equicontinuous subsets of
if
is a
sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
(such as a
pseudometrizable space) then this list may be extended to include:
- is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
at some (or equivalently, at every) point of its domain.
if
is
pseudometrizable or metrizable (such as a normed or
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
) then we may add to this list:
- is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
(that is, it maps bounded subsets of to bounded subsets of ).
if
is
seminormable space In mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some A ...
(such as a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
) then this list may be extended to include:
- maps some neighborhood of 0 to a bounded subset of
if
and
are both
normed or
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
s (with both seminorms denoted by
) then this list may be extended to include:
- for every there exists some such that
if
and
are Hausdorff locally convex spaces with
finite-dimensional then this list may be extended to include:
- the graph of is closed in
Continuity and boundedness
Throughout,
is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
between
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs).
Bounded on a set
The notion of "bounded set" for a topological vector space is that of being a
von Neumann bounded set.
If the space happens to also be a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
(or a
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
), such as the scalar field with the
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
for instance, then a subset
is von Neumann bounded if and only if it is
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
bounded; that is, if and only if
If
is a set then
is said to be if
is a
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
which if
is a normed (or seminormed) space happens if and only if
A linear map
is bounded on a set
if and only if it is bounded on
for every
(because
and any translation of a bounded set is again bounded).
Bounded linear maps
By definition, a linear map
between
TVSs is said to be and is called a if for every
(von Neumann) bounded subset of its domain,
is a bounded subset of it codomain; or said more briefly, if it is bounded on every bounded subset of its domain. When the domain
is a normed (or seminormed) space then it suffices to check this condition for the open or closed unit ball centered at the origin. Explicitly, if
denotes this ball then
is a bounded linear operator if and only if
is a bounded subset of
if
is also a (semi)normed space then this happens if and only if the
operator norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.
Introdu ...
is finite. Every
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
linear operator is bounded.
Bounded on a neighborhood and local boundedness
In contrast, a map
is said to be a point
or
if there exists a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of this point in
such that
is a
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
It is "" (of some point) if there exists point
in its domain at which it is locally bounded, in which case this linear map
is necessarily locally bounded at point of its domain.
The term "
" is sometimes used to refer to a map that is locally bounded at every point of its domain, but some functional analysis authors define "locally bounded" to instead be a synonym of "
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
", which are related but equivalent concepts. For this reason, this article will avoid the term "locally bounded" and instead say "locally bounded at every point" (there is no disagreement about the definition of "locally bounded ").
Bounded on a neighborhood implies continuous implies bounded
A linear map is "
bounded on a neighborhood" (of some point) if and only if it is locally bounded at every point of its domain, in which case it is necessarily
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(even if its domain is not a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
) and thus also
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
(because a continuous linear operator is always a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
).
For any linear map, if it is
bounded on a neighborhood then it is continuous, and if it is continuous then it is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
. The converse statements are not true in general but they are both true when the linear map's domain is a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
. Examples and additional details are now given below.
Continuous and bounded but not bounded on a neighborhood
The next example shows that it is possible for a linear map to be
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(and thus also bounded) but not bounded on any neighborhood. In particular, it demonstrates that being "bounded on a neighborhood" is always synonymous with being "
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
".
: If
is the identity map on some
locally convex topological vector space
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 vec ...
then this linear map is always continuous (indeed, even a
TVS-isomorphism
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is al ...
) and
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, but
is bounded on a neighborhood if and only if there exists a bounded neighborhood of the origin in
which
is equivalent to being a
seminormable space In mathematics, particularly in functional analysis, a seminorm is a Norm (mathematics), vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some A ...
(which if
is Hausdorff, is the same as being a
normable space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" ...
).
This shows that it is possible for a linear map to be continuous but bounded on any neighborhood.
Indeed, this example shows that every
locally convex space
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 vec ...
that is not seminormable has a linear TVS-
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
that is not bounded on any neighborhood of any point.
Thus although every linear map that is bounded on a neighborhood is necessarily continuous, the converse is not guaranteed in general.
Guaranteeing converses
To summarize the discussion below, for a linear map on a normed (or seminormed) space, being continuous, being
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
, and being bounded on a neighborhood are all
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
.
A linear map whose domain codomain is normable (or seminormable) is continuous if and only if it bounded on a neighborhood.
And a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
valued in a
locally convex space
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 vec ...
will be continuous if its domain is
(pseudo)metrizable or
bornological.
Guaranteeing that "continuous" implies "bounded on a neighborhood"
A TVS is said to be if there exists a neighborhood that is also a
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
. For example, every
normed or
seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
is a locally bounded TVS since the unit ball centered at the origin is a bounded neighborhood of the origin.
If
is a bounded neighborhood of the origin in a (locally bounded) TVS then its image under any continuous linear map will be a bounded set (so this map is thus bounded on this neighborhood
).
Consequently, a linear map from a locally bounded TVS into any other TVS is continuous if and only if it is
bounded on a neighborhood.
Moreover, any TVS with this property must be a locally bounded TVS. Explicitly, if
is a TVS such that every continuous linear map (into any TVS) whose domain is
is necessarily bounded on a neighborhood, then
must be a locally bounded TVS (because the
identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
is always a continuous linear map).
Any linear map from a TVS into a locally bounded TVS (such as any linear functional) is continuous if and only if it is bounded on a neighborhood.
Conversely, if
is a TVS such that every continuous linear map (from any TVS) with codomain
is necessarily
bounded on a neighborhood, then
must be a locally bounded TVS.
In particular, a linear functional on a arbitrary TVS is continuous if and only if it is bounded on a neighborhood.
Thus when the domain the codomain of a linear map is normable or seminormable, then continuity will be
equivalent
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
* Equivalence class (music)
*'' Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*''Equiva ...
to being bounded on a neighborhood.
Guaranteeing that "bounded" implies "continuous"
A continuous linear operator is always a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
.
But importantly, in the most general setting of a linear operator between arbitrary topological vector spaces, it is possible for a linear operator to be
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
but to be continuous.
A linear map whose domain is
pseudometrizable (such as any
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
) is
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
if and only if it is continuous.
The same is true of a linear map from a
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a ...
into a
locally convex space
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 vec ...
.
Guaranteeing that "bounded" implies "bounded on a neighborhood"
In general, without additional information about either the linear map or its domain or codomain, the map being "bounded" is not equivalent to it being "bounded on a neighborhood".
If
is a bounded linear operator from a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
into some TVS then
is necessarily continuous; this is because any open ball
centered at the origin in
is both a bounded subset (which implies that
is bounded since
is a bounded linear map) and a neighborhood of the origin in
so that
is thus bounded on this neighborhood
of the origin, which (as mentioned above) guarantees continuity.
Continuous linear functionals
Every linear functional on a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) is a linear operator so all of the properties described above for continuous linear operators apply to them.
However, because of their specialized nature, we can say even more about continuous linear functionals than we can about more general continuous linear operators.
Characterizing continuous linear functionals
Let
be a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) over the field
(
need not be
Hausdorff or
locally convex
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 ve ...
) and let
be a
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
on
The following are equivalent:
- is continuous.
- is uniformly continuous on
- is continuous at some point of
- is continuous at the origin.
* By definition, said to be continuous at the origin if for every open (or closed) ball of radius centered at in the codomain there exists some
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of the origin in such that If is a closed ball then the condition holds if and only if
** However, assuming that is instead an open ball, then is a sufficient but condition for to be true (consider for example when is the identity map on and ), whereas the non-strict inequality is instead a necessary but condition for to be true (consider for example and the closed neighborhood
- f is bounded on a neighborhood (of some point). Said differently, f is a locally bounded at some point of its domain.
* Explicitly, this means that there exists some neighborhood U of some point x \in X such that f(U) is a
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of \mathbb; that is, such that \displaystyle\sup_ , f(u), < \infty. This supremum over the neighborhood U is equal to 0 if and only if f = 0.
* Importantly, a linear functional being "bounded on a neighborhood" is in general equivalent to being a "bounded linear functional
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vector s ...
" because (as described above) it is possible for a linear map to be bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
but continuous. However, continuity and boundedness are equivalent if the domain is a normed or seminormed space In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...
; that is, for a linear functional on a normed space, being "bounded" is equivalent to being "bounded on a neighborhood".
- f is bounded on a neighborhood of the origin. Said differently, f is a locally bounded at the origin.
* The equality \sup_ , f(x), = , s, \sup_ , f(u), holds for all scalars s and when s \neq 0 then s U will be neighborhood of the origin. So in particular, if R := \displaystyle\sup_ , f(u), is a positive real number then for every positive real r > 0, the set N_r := \tfrac U is also a neighborhood of the origin and \displaystyle\sup_ , f(n), = r.
- There exists some neighborhood U of the origin such that \sup_ , f(u), \leq 1
* This inequality holds if and only if \sup_ , f(x), \leq r for every real r > 0, which shows that the positive scalar multiples \ of this single neighborhood U will satisfy the definition of continuity at the origin given in (4) above.
* By definition of the set U^, which is called the (absolute) polar of U, the inequality \sup_ , f(u), \leq 1 holds if and only if f \in U^. Polar sets, and thus also this particular inequality, play important roles in
duality theory
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the du ...
.
- f is a locally bounded at every point of its domain.
- The kernel of f is closed in X.
- Either f = 0 or else the kernel of f is dense in X.
- There exists a continuous seminorm p on X such that , f, \leq p.
* In particular, f is continuous if and only if the seminorm p := , f, is a continuous.
- The graph of f is closed.
- \operatorname f is continuous, where \operatorname f denotes the
real part
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
of f.
if
X and
Y are complex vector spaces then this list may be extended to include:
- The imaginary part of f is continuous.
if the domain
X is a
sequential space
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of count ...
then this list may be extended to include:
- f is
sequentially continuous
In topology and related fields of mathematics, a sequential space is a topological space whose topology can be completely characterized by its convergent/divergent sequences. They can be thought of as spaces that satisfy a very weak axiom of counta ...
at some (or equivalently, at every) point of its domain.
if the domain
X is
metrizable or pseudometrizable (for example, a
Fréchet space
In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces.
They are generalizations of Banach spaces (normed vector spaces that are complete with respect to the ...
or a
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
) then this list may be extended to include:
- f is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
(that is, it maps bounded subsets to bounded subsets).
if the domain
X is a
bornological space
In mathematics, particularly in functional analysis, a bornological space is a type of space which, in some sense, possesses the minimum amount of structure needed to address questions of boundedness of sets and linear maps, in the same way that a ...
(for example, a
pseudometrizable TVS) and
Y is
locally convex
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 ve ...
then this list may be extended to include:
- f is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
.
- f is sequentially continuous at some (or equivalently, at every) point of its domain.
- f is sequentially continuous at the origin.
and if in addition
X is a vector space over the
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
(which in particular, implies that
f is real-valued) then this list may be extended to include:
- There exists a continuous seminorm p on X such that f \leq p.
- For some real r, the half-space \ is closed.
- The above statement but with the word "some" replaced by "any."
Thus, if
X is a complex then either all three of
f, \operatorname f, and
\operatorname f are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(resp.
bounded
Boundedness or bounded may refer to:
Economics
* Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision
* Bounded e ...
), or else all three are
discontinuous
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all Function (mathematics), functions are Continuous function, continuous. If a function is not continuous at a point in its Domain of a function ...
(resp. unbounded).
Examples
Every linear map whose domain is a finite-dimensional Hausdorff
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) is continuous. This is not true if the finite-dimensional TVS is not Hausdorff.
Suppose
X is any Hausdorff TVS. Then
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
on
X is necessarily continuous if and only if every vector subspace of
X is closed. Every linear functional on
X is necessarily a bounded linear functional if and only if every
bounded subset
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
of
X is contained in a finite-dimensional vector subspace.
Properties
A
locally convex
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 ve ...
metrizable topological vector space
In functional analysis and related areas of mathematics, a metrizable (resp. pseudometrizable) topological vector space (TVS) is a TVS whose topology is induced by a metric (resp. pseudometric). An LM-space is an inductive limit of a sequence of ...
is
normable
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is ze ...
if and only if every bounded linear functional on it is continuous.
A continuous linear operator maps
bounded set
:''"Bounded" and "boundary" are distinct concepts; for the latter see boundary (topology). A circle in isolation is a boundaryless bounded set, while the half plane is unbounded yet has a boundary.
In mathematical analysis and related areas of mat ...
s into bounded sets.
The proof uses the facts that the translation of an open set in a linear topological space is again an open set, and the equality
F^(D) + x = F^(D + F(x))
for any subset
D of
Y and any
x \in X, which is true due to the
additivity
Additive may refer to:
Mathematics
* Additive function, a function in number theory
* Additive map, a function that preserves the addition operation
* Additive set-functionn see Sigma additivity
* Additive category, a preadditive category with fi ...
of
F.
Properties of continuous linear functionals
If
X is a complex
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
and
f is a linear functional on
X, then
\, f\, = \, \operatorname f\, (where in particular, one side is infinite if and only if the other side is infinite).
Every non-trivial continuous linear functional on a TVS
X is an
open map
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets.
That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y.
Likewise, a ...
.
Note that if
X is a real vector space,
f is a linear functional on
X, and
p is a seminorm on
X, then
, f, \leq p if and only if
f \leq p.
If
f : X \to \mathbb is a linear functional and
U \subseteq X is a non-empty subset, then by defining the sets
f(U) := \ \quad \text \quad , f(U), := \,
the supremum
\,\sup_ , f(u), \, can be written more succinctly as
\,\sup , f(U), \, because
\sup , f(U), ~=~ \sup \ ~=~ \sup_ , f(u), .
If
s is a scalar then
\sup , f(sU), ~=~ , s, \sup , f(U),
so that if
r > 0 is a real number and
\overline := \ is the closed ball of radius
r centered at the origin then
f(U) \subseteq \overline \quad \text \quad \sup , f(U), \leq 1 \quad \text \quad \sup , f(rU), \leq r \quad \text \quad f(rU) \subseteq \overline.
See also
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References
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{{Topological vector spaces
Theory of continuous functions
Functional analysis
Linear operators
Operator theory