Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.Precisely stated, the theorem says that if a space is either a complete metric space or a locally compact T2-space, then the intersection of every countable collection of dense open subsets of is necessarily dense in .The above-mentioned interplay with first and second category sets can be summarized by a single corollary, namely that spaces that are either complete metric spaces or locally compact Hausdorff spaces are of second category in themselves. To see that this follows from the above-stated theorem, let be either a complete metric..
Let be an involutive algebra over the field of complex numbers with involution . Then is a right Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of .
Let be an involutive algebra over the field of complex numbers with involution . Then is a left Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of . Left Hilbert algebras are historically known as generalized Hilbert algebras (Takesaki 1970).A basic result in functional analysis says that if the involution map on a left Hilbert algebra is an antilinear isometry with respect to the inner product , then is also a right Hilbert algebra with respect to the involution . The converse also holds.
The class of all regular sequences of particularly well-behaved functions equivalent to a given regular sequence. A distribution is sometimes also called a "generalized function" or "ideal function." As its name implies, a generalized function is a generalization of the concept of a function. For example, in physics, a baseball being hit by a bat encounters a force from the bat, as a function of time. Since the transfer of momentum from the bat is modeled as taking place at an instant, the force is not actually a function. Instead, it is a multiple of the delta function. The set of distributions contains functions (locally integrable) and Radon measures. Note that the term "distribution" is closely related to statistical distributions.Generalized functions are defined as continuous linear functionals over a space of infinitely differentiable functions such that all continuous functions have derivatives..
Let and be linear operators from domains and , respectively, into a Hilbert space . It is said that extends if and if for any vector .
A vector on a Hilbert space is said to be cyclic if there exists some bounded linear operator on so that the set of orbitsis dense in . In this case, the operator is said to be a cyclic operator.
A bounded linear operator on a Hilbert space is said to be cyclic if there exists some vector for which the set of orbitsis dense in . In this case, the vector is said to be a cyclic vector.
A linear operator from its domain into a Hilbert space is closed if for any sequence of vectors such that and as , it follows that and .
A linear operator from its domain into a Hilbert space is closable if it has a closed extension where . Closable operators are sometimes called preclosed (Takesaki 1970), and the extension of is sometimes called the closure of .
Given a subalgebra of the algebra of bounded linear transformations from a Hilbert space onto itself, the vector is a separating vector for if the only operator satisfying is the zero operator .
The Kubo-Martin-Schwinger (KMS) condition is a kind of boundary-value condition which naturally emerges in quantum statistical mechanics and related areas.Given a quantum system with finite dimensional Hilbert space , define the function as(1)where is the imaginary unit and where is the Hamiltonian, i.e., the sum of the kinetic energies of all the particles in plus the potential energy of the particles associated with . Next, for any real number , define the thermal equilibrium as(2)where denotes the matrix trace. From and , one can define the so-called equilibrium correlation function where(3)whereby the KMS boundary condition says that(4)In particular, this identity relates to the state the values of the analytic function on the boundary of the strip(5)where here, denotes the imaginary part of and denotes the signum function applied to .In various literature, the KMS boundary condition is stated in sometimes-different contexts...
A subset of a topological space is said to be of first category in if can be written as the countable union of subsets which are nowhere dense in , i.e., if is expressible as a unionwhere each subset is nowhere dense in . Informally, one thinks of a first category subset as a "small" subset of the host space and indeed, sets of first category are sometimes referred to as meager. Sets which are not of first category are of second category.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The rational numbers are of first category and the irrational numbers are of second category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the set of integers with the subset topology inherited from is (vacuously) of..
Let be an involutive algebra over the field of complex numbers with involution . Then is a modular Hilbert algebra if has an inner product and a one-parameter group of automorphisms on , , satisfying: 1. . 2. For all , is bounded (hence, continuous) on . 3. The linear span of products , , is a dense subalgebra of . 4. for all , . 5. . 6. . 7. is an entire function of on . 8. For every real number , the set is dense in . The group is called the group of modular automorphisms.Note that the definition of modular Hilbert algebras is closely related to that of generalized Hilbert algebras in that every modular Hilbert algebra is a generalized Hilbert algebra provided that it satisfies one additional condition, namely that the involution is closable as a linear operator on the real pre-Hilbert space . This relationship is due, in part, to the fact that the properties of both structures were at the core of Tomita's original exposition of what is today the heart of Tomita-Takesaki..
A vector space with a T2-space topology such that the operations of vector addition and scalar multiplication are continuous. The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a Hilbert space and a Banach space are topological vector spaces.The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the Banach space , one of the L-p-spaces, is used. But if one is interested in pointwise convergence, then no norm will suffice. Instead, for each define the seminormon the vector space of functions on . The seminorms define a topology, the smallest one in which the seminorms are continuous. So is equivalent to for all , i.e., pointwise convergence. In a similar way, it is possible to define a topology for which "convergence" means uniform convergence on compact sets...
Let be a Hilbert space, the set of bounded linear operators from to itself, an operator on , and the operator spectrum of . Then if and is normal, there exists a unique resolution of the identity on the Borel subsets of which satisfiesFurthermore, every projection commutes with every that commutes with .
A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .A projection matrix is orthogonal iff(1)where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so(2)An example of a nonsymmetric projection matrix is(3)which projects onto the line .The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies(4)where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless ,(5)Let be a -algebra. An element..
There are at least two distinct (though related) notions of the term Hilbert algebrain functional analysis.In some literature, a linear manifold of a (not necessarily separable) Hilbert space is a Hilbert algebra if the following conditions are satisfied: 1. is dense in . 2. is a ring so that, for any , there is defined an element such that , , , and for any complex number . 3. For any , there exists an adjoint element such that , and . 4. For any , there exists a positive number such that for all . 5. For every , there exists a unique bounded linear operator on such that for all . Moreover, if for an element and for all , then . At least one author defines a Hilbert algebra to be a quasi-Hilbertalgebrafor which for all (Dixmier 1981).
The group algebra , where is a field and a group with the operation , is the set of all linear combinations of finitely many elements of with coefficients in , hence of all elements of the form(1)where and for all . This element can be denoted in general by(2)where it is assumed that for all but finitely many elements of . is an algebra over with respect to the addition defined by the rule(3)the product by a scalar given by(4)and the multiplication(5)From this definition, it follows that the identity element of is the unit of , and that is commutative iff is an Abelian group.If the field is replaced by a unit ring , the addition and the multiplication defined above yield the group ring .If , and is the usual addition of integers, the group ring is isomorphic to the ring formed by all sums(6)where are integers, and for all indices .Let be a locally compact group and be a left invariant Haar measure on . Then the Banach space under the product given by the convolution..