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Page "Banach algebra" ¶ 27
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Every and unital
** Every unital ring other than the trivial ring contains a maximal ideal.
* Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers.
* Every commutative real unital Noetherian Banach algebra ( possibly having zero divisors ) is finite-dimensional.

Every and real
* Every real Banach algebra which is a division algebra is isomorphic to the reals, the complexes, or the quaternions.
Every sequence that ran off to infinity in the real line will then converge toin this compactification.
Every real number, whether integer, rational, or irrational, has a unique location on the line.
Every real number has a ( possibly infinite ) decimal representation ; i. e., it can be written as
Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.
Every ordered field is a formally real field.
Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.
* Every separable metric space is isometric to a subset of the ( non-separable ) Banach space l < sup >∞</ sup > of all bounded real sequences with the supremum norm ; this is known as the Fréchet embedding.
Every non-negative real number a has a unique non-negative square root, called the principal square root, which is denoted by, where √ is called the radical sign or radix.
Every dual number has the form z = a + bε with a and b uniquely determined real numbers.
Every real number has an additive inverse ( i. e. an inverse with respect to addition ) given by.
Every nonzero real number has a multiplicative inverse ( i. e. an inverse with respect to multiplication ) given by ( or ).
Every real number, rational or not, is equated to one and only one cut of rationals.
Every sedenion is a real linear combination of the unit sedenions 1, < var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,
Every octonion is a real linear combination of the unit octonions:
Every real symmetric matrix is Hermitian, and therefore all its eigenvalues are real.
Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.
In his book Nirvana: The Stories Behind Every Song, Chuck Crisafulli writes that the song " stands out in the Cobain canon as a song with a very specific genesis and a very real subject ".
* Every real number greater than zero or every complex number except 0 has two square roots.
Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.

Every and Banach
* Theorem Every reflexive normed space is a Banach space.
Every Hilbert space X is a Banach space because, by definition, a Hilbert space is complete with respect to the norm associated with its inner product, where a norm and an inner product are said to be associated if for all x ∈ X.
Every normed vector space V sits as a dense subspace inside a Banach space ; this Banach space is essentially uniquely defined by V and is called the completion of V.
* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

Every and algebra
Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions.
Every Boolean algebra ( A, ∧, ∨) gives rise to a ring ( A, +, ·) by defining a + b := ( a ∧ ¬ b ) ∨ ( b ∧ ¬ a ) = ( a ∨ b ) ∧ ¬( a ∧ b ) ( this operation is called symmetric difference in the case of sets and XOR in the case of logic ) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity element of the ring is the 1 of the Boolean algebra.
Every division ring is therefore a division algebra over its center.
Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).
Every vector v in determines a linear map from R to taking 1 to v, which can be thought of as a Lie algebra homomorphism.
Every associative algebra is obviously power-associative, but so are all other alternative algebras ( like the octonions, which are non-associative ) and even some non-alternative algebras like the sedenions.
Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.
* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.
Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.
Every state on a C *- algebra is of the above type.
Every Boolean algebra is a Heyting algebra when a → b is defined as usual as ¬ a ∨ b, as is every complete distributive lattice when a → b is taken to be the supremum of the set of all c for which a ∧ c ≤ b. The open sets of a topological space form a complete distributive lattice and hence a Heyting algebra.
Every Heyting algebra with exactly one coatom is subdirectly irreducible, whence every Heyting algebra can be made an SI by adjoining a new top.
* Every Boolean algebra is a Heyting algebra, with given by.

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