In
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 ...
, more specifically
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
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
, the notion of unbounded operator provides an abstract framework for dealing with
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s, unbounded
observable
In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum ph ...
s in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, and other cases.
The term "unbounded operator" can be misleading, since
* "unbounded" should sometimes be understood as "not necessarily bounded";
* "operator" should be understood as "
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 ...
" (as in the case of "bounded operator");
* the domain of the operator is a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...
, not necessarily the whole space;
* this linear subspace is not necessarily
closed; often (but not always) it is assumed to be
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
;
* in the special case of a bounded operator, still, the domain is usually assumed to be the whole space.
In contrast to
bounded 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 vector s ...
s, unbounded operators on a given space do not form an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
, nor even a linear space, because each one is defined on its own domain.
The term "operator" often means "bounded linear operator", but in the context of this article it means "unbounded operator", with the reservations made above. The given space is assumed to be a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
. Some generalizations to
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 ...
s and more general
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 are possible.
Short history
The theory of unbounded operators developed in the late 1920s and early 1930s as part of developing a rigorous mathematical framework for
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The theory's development is due to
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
and
Marshall Stone
Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras.
Biography
Stone was the son of Harlan Fiske Stone, who wa ...
.
Von Neumann introduced using
graphs
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
to analyze unbounded operators in 1932.
Definitions and basic properties
Let be
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 ...
s. An unbounded operator (or simply ''operator'') 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 ...
from a linear subspace —the domain of —to the space .
Contrary to the usual convention, may not be defined on the whole space .
An operator is said to be
closed if its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
.
(Here, the graph is a linear subspace of the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
, defined as the set of all pairs , where runs over the domain of .) Explicitly, this means that for every sequence of points from the domain of such that and , it holds that belongs to the domain of and .
The closedness can also be formulated in terms of the ''graph norm'': an operator is closed if and only if its domain is a
complete space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in .
Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
with respect to the norm:
:
An operator is said to be
densely defined if its domain is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in .
This also includes operators defined on the entire space , since the whole space is dense in itself. The denseness of the domain is necessary and sufficient for the existence of the adjoint (if and are Hilbert spaces) and the transpose; see the sections below.
If is closed, densely defined and
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 ...
on its domain, then its domain is all of .
A densely defined operator on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is called bounded from below if is a positive operator for some real number . That is, for all in the domain of (or alternatively since is arbitrary).
If both and are bounded from below then is bounded.
Example
Let denote the space of continuous functions on the unit interval, and let denote the space of continuously differentiable functions. We equip
with the supremum norm,
, making it a Banach space. Define the classical differentiation operator by the usual formula:
:
Every differentiable function is continuous, so . We claim that is a well-defined unbounded operator, with domain . For this, we need to show that
is linear and then, for example, exhibit some
such that
and
.
This is a linear operator, since a linear combination of two continuously differentiable functions is also continuously differentiable, and
:
The operator is not bounded. For example,
:
satisfy
:
but
:
as
.
The operator is densely defined, and closed.
The same operator can be treated as an operator for many choices of Banach space and not be bounded between any of them. At the same time, it can be bounded as an operator for other pairs of Banach spaces , and also as operator for some topological vector spaces . As an example let be an open interval and consider
:
where:
:
Adjoint
The adjoint of an unbounded operator can be defined in two equivalent ways. Let
be an unbounded operator between Hilbert spaces.
First, it can be defined in a way analogous to how one defines the adjoint of a bounded operator. Namely, the adjoint
of is defined as an operator with the property:
More precisely,
is defined in the following way. If
is such that
is a continuous linear functional on the domain of , then
is declared to be an element of
and after extending the linear functional to the whole space via the
Hahn–Banach theorem
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
, it is possible to find some
in
such that
since
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the R ...
allows the continuous dual of the Hilbert space
to be identified with the set of linear functionals given by the inner product. This vector
is uniquely determined by
if and only if the linear functional
is densely defined; or equivalently, if is densely defined. Finally, letting
completes the construction of
which is necessarily a linear map. The adjoint
exists if and only if is densely defined.
By definition, the domain of
consists of elements
in
such that
is continuous on the domain of . Consequently, the domain of
could be anything; it could be trivial (that is, contains only zero).
It may happen that the domain of
is a closed
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
and
vanishes everywhere on the domain.
Thus, boundedness of
on its domain does not imply boundedness of . On the other hand, if
is defined on the whole space then is bounded on its domain and therefore can be extended by continuity to a bounded operator on the whole space. If the domain of
is dense, then it has its adjoint
A closed densely defined operator is bounded if and only if
is bounded.
The other equivalent definition of the adjoint can be obtained by noticing a general fact. Define a linear operator
as follows:
Since
is an isometric surjection, it is unitary. Hence:
is the graph of some operator
if and only if is densely defined.
A simple calculation shows that this "some"
satisfies:
for every in the domain of . Thus
is the adjoint of .
It follows immediately from the above definition that the adjoint
is closed.
In particular, a self-adjoint operator (meaning
) is closed. An operator is closed and densely defined if and only if
Some well-known properties for bounded operators generalize to closed densely defined operators. The kernel of a closed operator is closed. Moreover, the kernel of a closed densely defined operator
coincides with the orthogonal complement of the range of the adjoint. That is,
von Neumann's theorem In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces.
Statement of the theorem
Let G and H be Hilbert spaces, and let T : \operatorname(T) \subseteq G \to H be an unbounded operator from G ...
states that
and
are self-adjoint, and that
and
both have bounded inverses. If
has trivial kernel, has dense range (by the above identity.) Moreover:
: is surjective if and only if there is a
such that
for all
in
(This is essentially a variant of the so-called
closed range theorem
In the mathematical theory of Banach spaces, the closed range theorem gives necessary and sufficient conditions for a closed densely defined operator to have closed range.
History
The theorem was proved by Stefan Banach in his 1932 '' Théori ...
.) In particular, has closed range if and only if
has closed range.
In contrast to the bounded case, it is not necessary that
since, for example, it is even possible that
does not exist. This is, however, the case if, for example, is bounded.
A densely defined, closed operator is called ''
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
'' if it satisfies the following equivalent conditions:
*
;
* the domain of is equal to the domain of
and
for every in this domain;
* there exist self-adjoint operators
such that
and
for every in the domain of .
Every self-adjoint operator is normal.
Transpose
Let
be an operator between Banach spaces. Then the ''
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 ...
'' (or ''dual'')
of
is the linear operator satisfying:
for all
and
Here, we used the notation:
The necessary and sufficient condition for the transpose of
to exist is that
is densely defined (for essentially the same reason as to adjoints, as discussed above.)
For any Hilbert space
there is the anti-linear isomorphism:
given by
where
Through this isomorphism, the transpose
relates to the adjoint
in the following way:
where
. (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.
Closed linear operators
Closed linear operators are a class of
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 ...
s on
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 ...
s. They are more general than
bounded 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 vector s ...
s, and therefore not 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 ...
, but they still retain nice enough properties that one can define the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
and (with certain assumptions)
functional calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral the ...
for such operators. Many important linear operators which fail to be bounded turn out to be closed, such as the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
and a large class of
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
s.
Let be two
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 ...
s. 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 ...
is closed if for every
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
in
converging to in such that as one has and .
Equivalently, is closed if its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is
closed in the
direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
.
Given a linear operator , not necessarily closed, if the closure of its graph in happens to be the graph of some operator, that operator is called the closure of , and we say that is closable. Denote the closure of by . It follows that is the
restriction
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
of to .
A core (or essential domain) of a closable operator is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of such that the closure of the restriction of to is .
Example
Consider the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
operator where is the Banach space of all
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s on an
interval .
If one takes its domain to be , then is a closed operator which is not bounded.
On the other hand if , then will no longer be closed, but it will be closable, with the closure being its extension defined on .
Symmetric operators and self-adjoint operators
An operator ''T'' on a Hilbert space is ''symmetric'' if and only if for each ''x'' and ''y'' in the domain of we have
. A densely defined operator is symmetric if and only if it agrees with its adjoint ''T''
∗ restricted to the domain of ''T'', in other words when ''T''
∗ is an extension of .
In general, if ''T'' is densely defined and symmetric, the domain of the adjoint ''T''
∗ need not equal the domain of ''T''. If ''T'' is symmetric and the domain of ''T'' and the domain of the adjoint coincide, then we say that ''T'' is ''self-adjoint''. Note that, when ''T'' is self-adjoint, the existence of the adjoint implies that ''T'' is densely defined and since ''T''
∗ is necessarily closed, ''T'' is closed.
A densely defined operator ''T'' is ''symmetric'', if the subspace (defined in a previous section) is orthogonal to its image under ''J'' (where ''J''(''x'',''y''):=(''y'',-''x'')).
[Follows from and the definition via adjoint operators.]
Equivalently, an operator ''T'' is ''self-adjoint'' if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators , are surjective, that is, map the domain of ''T'' onto the whole space ''H''. In other words: for every ''x'' in ''H'' there exist ''y'' and ''z'' in the domain of ''T'' such that and .
An operator ''T'' is ''self-adjoint'', if the two subspaces , are orthogonal and their sum is the whole space
This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators.
A symmetric operator is often studied via its
Cayley transform In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform ...
.
An operator ''T'' on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number
is real for all ''x'' in the domain of ''T''.
A densely defined closed symmetric operator ''T'' is self-adjoint if and only if ''T''
∗ is symmetric.
It may happen that it is not.
A densely defined operator ''T'' is called ''positive''
(or ''nonnegative''
) if its quadratic form is nonnegative, that is,
for all ''x'' in the domain of ''T''. Such operator is necessarily symmetric.
The operator ''T''
∗''T'' is self-adjoint
and positive
for every densely defined, closed ''T''.
The
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
applies to self-adjoint operators
and moreover, to normal operators,
but not to densely defined, closed operators in general, since in this case the spectrum can be empty.
A symmetric operator defined everywhere is closed, therefore bounded,
which is the
Hellinger–Toeplitz theorem In functional analysis, a branch of mathematics, the Hellinger–Toeplitz theorem states that an everywhere-defined symmetric operator on a Hilbert space with inner product \langle \cdot , \cdot \rangle is bounded. By definition, an operator ' ...
.
Extension-related
By definition, an operator ''T'' is an ''extension'' of an operator ''S'' if .
An equivalent direct definition: for every ''x'' in the domain of ''S'', ''x'' belongs to the domain of ''T'' and .
Note that an everywhere defined extension exists for every operator, which is a purely algebraic fact explained at
Discontinuous linear map#General existence theorem and based on the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
. If the given operator is not bounded then the extension is a
discontinuous linear map In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved ar ...
. It is of little use since it cannot preserve important properties of the given operator (see below), and usually is highly non-unique.
An operator ''T'' is called ''closable'' if it satisfies the following equivalent conditions:
* ''T'' has a closed extension;
* the closure of the graph of ''T'' is the graph of some operator;
* for every sequence (''x
n'') of points from the domain of ''T'' such that ''x
n'' → 0 and also ''Tx
n'' → ''y'' it holds that .
Not all operators are closable.
A closable operator ''T'' has the least closed extension
called the ''closure'' of ''T''. The closure of the graph of ''T'' is equal to the graph of
Other, non-minimal closed extensions may exist.
A densely defined operator ''T'' is closable if and only if ''T''
∗ is densely defined. In this case
and
If ''S'' is densely defined and ''T'' is an extension of ''S'' then ''S''
∗ is an extension of ''T''
∗.
Every symmetric operator is closable.
A symmetric operator is called ''maximal symmetric'' if it has no symmetric extensions, except for itself.
Every self-adjoint operator is maximal symmetric.
The converse is wrong.
An operator is called ''essentially self-adjoint'' if its closure is self-adjoint.
An operator is essentially self-adjoint if and only if it has one and only one self-adjoint extension.
A symmetric operator may have more than one self-adjoint extension, and even a continuum of them.
A densely defined, symmetric operator ''T'' is essentially self-adjoint if and only if both operators , have dense range.
Let ''T'' be a densely defined operator. Denoting the relation "''T'' is an extension of ''S''" by ''S'' ⊂ ''T'' (a conventional abbreviation for Γ(''S'') ⊆ Γ(''T'')) one has the following.
* If ''T'' is symmetric then ''T'' ⊂ ''T''
∗∗ ⊂ ''T''
∗.
* If ''T'' is closed and symmetric then ''T'' = ''T''
∗∗ ⊂ ''T''
∗.
* If ''T'' is self-adjoint then ''T'' = ''T''
∗∗ = ''T''
∗.
* If ''T'' is essentially self-adjoint then ''T'' ⊂ ''T''
∗∗ = ''T''
∗.
Importance of self-adjoint operators
The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
holds for self-adjoint operators. In combination with
Stone's theorem on one-parameter unitary groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families
:(U_)_
o ...
it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see
Self-adjoint operator#Self-adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing
time evolution
Time evolution is the change of state brought about by the passage of time, applicable to systems with internal state (also called ''stateful systems''). In this formulation, ''time'' is not required to be a continuous parameter, but may be disc ...
in classical and quantum mechanics.
See also
*
Hilbert space#Unbounded operators
*
Stone–von Neumann theorem
*
Bounded 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 vector s ...
Notes
References
* (see Chapter 12 "General theory of unbounded operators in Hilbert spaces").
*
*
*
*
* (see Chapter 5 "Unbounded operators").
* (see Chapter 8 "Unbounded operators").
*
*
{{DEFAULTSORT:Unbounded Operator
Linear operators
Operator theory
de:Linearer Operator#Unbeschränkte lineare Operatoren