Locally convex spaces, the hyperplane separation theorem, and the
... K is said to be extreme if it is nonempty convex and furthermore if any x ∈ A is a convex combination of two points x1 and x2 in K then we must have x1 , x2 ∈ A. It is clear that if a family of extreme subsets of K has nonempty intersection then this intersection is also extreme. In particular K is ...
... K is said to be extreme if it is nonempty convex and furthermore if any x ∈ A is a convex combination of two points x1 and x2 in K then we must have x1 , x2 ∈ A. It is clear that if a family of extreme subsets of K has nonempty intersection then this intersection is also extreme. In particular K is ...
C. Foias, S. Hamid, C. Onica, and C. Pearcy
... is to use the little known but useful main theorem from [6] to make some progress on this old outstanding problem. For the reader’s convenience, we first state the main result from [6], which characterizes the set of all (positive semidefinite) Q that arise from the Aronszajn-Smith “invariant subspa ...
... is to use the little known but useful main theorem from [6] to make some progress on this old outstanding problem. For the reader’s convenience, we first state the main result from [6], which characterizes the set of all (positive semidefinite) Q that arise from the Aronszajn-Smith “invariant subspa ...
Inf-sup conditions
... also implies that the range R(A0 ) is closed and thus form a linear subspace of U0 . Choosing a convergent sequence {A0 vk }, by (2), we know {vk } is also a Cauchy sequence and thus converges to some v ∈ V. The continuity of A0 shows that A0 vk converges to A0 v and thus R(A0 ) is closed. We can th ...
... also implies that the range R(A0 ) is closed and thus form a linear subspace of U0 . Choosing a convergent sequence {A0 vk }, by (2), we know {vk } is also a Cauchy sequence and thus converges to some v ∈ V. The continuity of A0 shows that A0 vk converges to A0 v and thus R(A0 ) is closed. We can th ...
The music of the primes, harmonic music noise between red and
... The Berry-Keating conjecture [BeM] is about some evidence that the eigenvalues are energy levels, that is eigenvalues of a Hermitian quantum (“Riemann”) operator associated with the classical Hamiltonian H(x,p)=xp, where x is the (one-dimensional) position coordinate and p the conjugate momentum. “T ...
... The Berry-Keating conjecture [BeM] is about some evidence that the eigenvalues are energy levels, that is eigenvalues of a Hermitian quantum (“Riemann”) operator associated with the classical Hamiltonian H(x,p)=xp, where x is the (one-dimensional) position coordinate and p the conjugate momentum. “T ...
On the Choquet-Dolecki Theorem
... Lemma 1.3. Let Φ be a multifunction from a topological space T into a Hausdorff space X. If {Un : n ∈ N} is a local base for t0 ∈ T and Φ is usc at t0 , then each sequence (xn : n ∈ N) in X with xn ∈ Φ(Un )\Φ(t0 ) has a cluster point in Φ(t0 ). In particular, if x = lim xn and xn ∈ Φ(Un )\Φ(t0 ) n→∞ ...
... Lemma 1.3. Let Φ be a multifunction from a topological space T into a Hausdorff space X. If {Un : n ∈ N} is a local base for t0 ∈ T and Φ is usc at t0 , then each sequence (xn : n ∈ N) in X with xn ∈ Φ(Un )\Φ(t0 ) has a cluster point in Φ(t0 ). In particular, if x = lim xn and xn ∈ Φ(Un )\Φ(t0 ) n→∞ ...
MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES
... MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES Abstract. This is an explanatory note on what the basic definitions of linear algebra mean when the vector spaces are infinite-dimensional. ...
... MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES Abstract. This is an explanatory note on what the basic definitions of linear algebra mean when the vector spaces are infinite-dimensional. ...
MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES
... MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES Abstract. This is an explanatory note on what the basic definitions of linear algebra mean when the vector spaces are infinite-dimensional. ...
... MATH 412: NOTE ON INFINITE-DIMENSIONAL VECTOR SPACES Abstract. This is an explanatory note on what the basic definitions of linear algebra mean when the vector spaces are infinite-dimensional. ...
+ y - U.I.U.C. Math
... the cases specified by (5.2) and (5.3), and thus establish the corresponding cases of (5.1). To do this, we generalize the argument of §4; here Clarkson's inequalities will replace the equality ...
... the cases specified by (5.2) and (5.3), and thus establish the corresponding cases of (5.1). To do this, we generalize the argument of §4; here Clarkson's inequalities will replace the equality ...
1. FINITE-DIMENSIONAL VECTOR SPACES
... By now you’ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices, but at other times the abstract approach allows us more freedom. Recall that a field is a mathematical system having two operations + a ...
... By now you’ll have acquired a fair knowledge of matrices. These are a concrete embodiment of something rather more abstract. Sometimes it is easier to use matrices, but at other times the abstract approach allows us more freedom. Recall that a field is a mathematical system having two operations + a ...
Physical justification for using the tensor product to describe two
... weakly modular lattice L of its yes-no experiments. The next step in the study of the system is to investigate which yes-no experiments are true at a certain moment. These are indeed the properties which are elements of reality for the system at that moment: they represent the state of the system. T ...
... weakly modular lattice L of its yes-no experiments. The next step in the study of the system is to investigate which yes-no experiments are true at a certain moment. These are indeed the properties which are elements of reality for the system at that moment: they represent the state of the system. T ...
1 Complex Numbers in Quantum Mechanics
... in a complex vector space, possibly one of an infinite number of dimensions. Call the vector u. Now suppose there is another state of our system, described by a different vector v. Generalizing the discussion in the photon case, it is necessary that these both be unit vectors, so our space needs to ha ...
... in a complex vector space, possibly one of an infinite number of dimensions. Call the vector u. Now suppose there is another state of our system, described by a different vector v. Generalizing the discussion in the photon case, it is necessary that these both be unit vectors, so our space needs to ha ...
Relatives of the quotient of the complex projective plane by complex
... singular points on the cones of the degenerate points are equal, in the real, complex and quaternionic cases, to 2, 3 and 5 (these numbers are the codimensions of the onedimensional spaces of the diagonal forms of two variables in the spaces of quadratic, Hermitian and hyperhermitian forms of two va ...
... singular points on the cones of the degenerate points are equal, in the real, complex and quaternionic cases, to 2, 3 and 5 (these numbers are the codimensions of the onedimensional spaces of the diagonal forms of two variables in the spaces of quadratic, Hermitian and hyperhermitian forms of two va ...
Algebraic topology and operators in Hilbert space
... locally constant) functions of x: they are only semi-continuous, that is dim Ker F(x ) ...
... locally constant) functions of x: they are only semi-continuous, that is dim Ker F(x ) ...
On compact operators - NC State: WWW4 Server
... We have already noted that finite-dimensional operators on normed linear spaces are compact. Moreover, we know by Theorem 2.6 that the limit (in the operator norm) of a sequence of finite-dimensional operators is a compact operator. Moreover, we have seen that the range of compact operators can be a ...
... We have already noted that finite-dimensional operators on normed linear spaces are compact. Moreover, we know by Theorem 2.6 that the limit (in the operator norm) of a sequence of finite-dimensional operators is a compact operator. Moreover, we have seen that the range of compact operators can be a ...
Explicit building the nonlinear coherent states associated to weighted shift Zp dp+1/ dzp+1 of order p in classical Bargmann representation
... p+1 ; p = 0, 1, ..... are non-wandering and hypercyclic operators on classical Bargmann space, the space of entire functions with Gaussian measure.In this way,the aim of this paper is to construct nonlinear coherent states corresponding to Hp , where A and A∗ are the standard Boson annihilation and ...
... p+1 ; p = 0, 1, ..... are non-wandering and hypercyclic operators on classical Bargmann space, the space of entire functions with Gaussian measure.In this way,the aim of this paper is to construct nonlinear coherent states corresponding to Hp , where A and A∗ are the standard Boson annihilation and ...
THE C∗-ALGEBRAIC FORMALISM OF QUANTUM MECHANICS
... 2. A Brief Look at Classical Mechanics In order to motivate more natural axioms of a quantum theory (as mentioned in the abstract), I first wish to examine (superficially) the mathematical formulation of classical mechanics (in the Hamiltonian sense). In any theory of mechanics, we must come to grip ...
... 2. A Brief Look at Classical Mechanics In order to motivate more natural axioms of a quantum theory (as mentioned in the abstract), I first wish to examine (superficially) the mathematical formulation of classical mechanics (in the Hamiltonian sense). In any theory of mechanics, we must come to grip ...
Quantum Probability Theory
... In the last ten years or so a succesful strategy has become popular in mathematics. The most widely known example of this strategy is no doubt non-commutative geometry, as explained in the imaginative book of Alain Connes (1990,1994). Nowadays we have non-commutative topological spaces, quantum grou ...
... In the last ten years or so a succesful strategy has become popular in mathematics. The most widely known example of this strategy is no doubt non-commutative geometry, as explained in the imaginative book of Alain Connes (1990,1994). Nowadays we have non-commutative topological spaces, quantum grou ...
chapter 10. relation to quantum mechanics
... Objectivity is a property of a class of experimenters on the system; it expresses the mutual consistency of descriptions of the system by the various experimenters in the class. At this level of analysis, the group J is associated to the class of experimenters; one does not need to have a “configura ...
... Objectivity is a property of a class of experimenters on the system; it expresses the mutual consistency of descriptions of the system by the various experimenters in the class. At this level of analysis, the group J is associated to the class of experimenters; one does not need to have a “configura ...
Lecture 3
... According to the above we use the pair of numbers ( x, y) to specify a complex number. Geometrically we could also use the length and angle variables in polar coordinates (q, r ). Theta is calle the phase. We know that r = x 2 + y2 , ...
... According to the above we use the pair of numbers ( x, y) to specify a complex number. Geometrically we could also use the length and angle variables in polar coordinates (q, r ). Theta is calle the phase. We know that r = x 2 + y2 , ...
Linear Space - El Camino College
... Some examples of common elements of linear spaces include traditional ndimensional vectors, functions of a real variable, convergent infinite series, matrices, and sequences of real numbers. Most often, the operations of addition (of elements in the linear space) and multiplication (of the elements ...
... Some examples of common elements of linear spaces include traditional ndimensional vectors, functions of a real variable, convergent infinite series, matrices, and sequences of real numbers. Most often, the operations of addition (of elements in the linear space) and multiplication (of the elements ...
VECtoR sPACEs We first define the notion of a field, examples of
... This right R-vector space is 2-dimensional. (4) The set C of complex numbers also admits a structure of right Q-vector space with sum and scalar multiplication given by the same formulas as in (3), but where we now only allow a ∈ Q. The dimension of the resulting right Q-vector space is equal to the ...
... This right R-vector space is 2-dimensional. (4) The set C of complex numbers also admits a structure of right Q-vector space with sum and scalar multiplication given by the same formulas as in (3), but where we now only allow a ∈ Q. The dimension of the resulting right Q-vector space is equal to the ...
Titles and Abstracts
... systems with six levels. Using numerical, computer-algebraic and analytic methods, various partial results have been obtained all of which are compatible with the conjecture that no more than three MU bases exist. I will emphasise the case of MU bases consisting of product states only for which stro ...
... systems with six levels. Using numerical, computer-algebraic and analytic methods, various partial results have been obtained all of which are compatible with the conjecture that no more than three MU bases exist. I will emphasise the case of MU bases consisting of product states only for which stro ...
GANTMACHER-KRE˘IN THEOREM FOR 2 NONNEGATIVE OPERATORS IN SPACES OF FUNCTIONS
... it was proved that there exists a converging-to-zero sequence of positive simple eigenvalues λ1 > λ2 > · · · > λn > · · · with eigenfunctions en (t) that has exactly n − 1 changes of sign, corresponding to the nth eigenvalue λn (see [3, page 211]). In connection with the formulated Gantmacher-Kreı̆n ...
... it was proved that there exists a converging-to-zero sequence of positive simple eigenvalues λ1 > λ2 > · · · > λn > · · · with eigenfunctions en (t) that has exactly n − 1 changes of sign, corresponding to the nth eigenvalue λn (see [3, page 211]). In connection with the formulated Gantmacher-Kreı̆n ...
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer)—and ergodic theory, which forms the mathematical underpinning of thermodynamics. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of ""dropping the altitude"" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.