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51 relations: Algebra over a field, Banach algebra, Banach space, Bijection, Bound state, Bounded inverse theorem, Bounded operator, Bounded set, Closed graph theorem, Closed set, Compact operator, Complement (set theory), Complex number, Complex plane, Decomposition of spectrum (functional analysis), Densely defined operator, Dimension (vector space), Eigenvalues and eigenvectors, Empty set, Essential spectrum, Functional analysis, Hilbert space, Holomorphic function, Hydrogen atom, Identity function, Inverse function, Ionization, Isometry, Liouville's theorem (complex analysis), Lp space, Mathematical formulation of quantum mechanics, Mathematics, Matrix (mathematics), Molecular Hamiltonian, Multiplication operator, Neumann series, Normal operator, Pseudospectrum, Quantum mechanics, Resolvent formalism, Resolvent set, Rydberg formula, Self-adjoint operator, Shift operator, Spectral radius, Spectral theorem, Spectral theory, Spectral theory of ordinary differential equations, Unbounded operator, Unit (ring theory), ..., Vector space. Expand index (1 more) »
Algebra over a field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product.
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Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach space, i.e. a normed space and complete in the metric induced by the norm.
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Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.
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Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
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Bound state
In quantum physics, a bound state is a special quantum state of a particle subject to a potential such that the particle has a tendency to remain localised in one or more regions of space.
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Bounded inverse theorem
In mathematics, the bounded inverse theorem (or inverse mapping theorem) is a result in the theory of bounded linear operators on Banach spaces.
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Bounded operator
In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).
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Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.
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Closed graph theorem
In mathematics, the closed graph theorem is a basic result which characterizes continuous functions in terms of their graphs.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset (has compact closure) of Y. Such an operator is necessarily a bounded operator, and so continuous.
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Complement (set theory)
In set theory, the complement of a set refers to elements not in.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Decomposition of spectrum (functional analysis)
The spectrum of a linear operator T that operates on a Banach space X (a fundamental concept of functional analysis) consists of all scalars \lambda such that the operator T-\lambda does not have a bounded inverse on X. The spectrum has a standard decomposition into three parts.
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Densely defined operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.
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Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Empty set
In mathematics, and more specifically set theory, the empty set or null set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.
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Essential spectrum
In mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible".
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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, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
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Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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Inverse function
In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.
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Ionization
Ionization or ionisation, is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons to form ions, often in conjunction with other chemical changes.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Liouville's theorem (complex analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville, states that every bounded entire function must be constant.
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Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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Mathematical formulation of quantum mechanics
The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Molecular Hamiltonian
In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule.
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Multiplication operator
In operator theory, a multiplication operator is an operator defined on some vector space of functions and whose value at a function is given by multiplication by a fixed function.
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Neumann series
A Neumann series is a mathematical series of the form where T is an operator.
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Normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN*.
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Pseudospectrum
In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues.
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
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Resolvent formalism
In mathematics, the resolvent formalism is a technique for applying concepts from complex analysis to the study of the spectrum of operators on Banach spaces and more general spaces.
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Resolvent set
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved".
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Rydberg formula
The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements.
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Self-adjoint operator
In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.
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Shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function to its translation.
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Spectral radius
In mathematics, the spectral radius of a square matrix or a bounded linear operator is the largest absolute value of its eigenvalues (i.e. supremum among the absolute values of the elements in its spectrum).
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Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
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Spectral theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces.
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Spectral theory of ordinary differential equations
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation.
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Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Redirects here:
Approximate eigenvalue, Compression spectrum, Operator spectrum, Point spectrum, Spectral representation, Spectrum of an operator.
References
[1] https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)
