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Page "Continuous linear extension" ¶ 2
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Every and bounded
* Pseudocompact: Every real-valued continuous function on the space is bounded.
* 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 finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.
* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.
* Every continuous function f: → R is bounded.
This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.
For example, to study the theorem “ Every bounded sequence of real numbers has a supremum ” it is necessary to use a base system which can speak of real numbers and sequences of real numbers.
Every maximal outerplanar graph with n vertices has exactly 2n − 3 edges, and every bounded face of a maximal outerplanar graph is a triangle.
Every three years the Company makes an award to the three buildings or structures, in the area bounded by the M25 motorway, which respectively embody the most outstanding example of brickwork, of slated or tiled roof and of hard-surface tiled wall and / or floor.
Every Polish town was bounded to put up a quantity of soldiers-this was a conspicuous sign of a power of a given town how much soldiers it had to put up.
Every complete lattice is also a bounded lattice, which is to say that it has a greatest and least element.
Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
Every subset of a totally bounded space is a totally bounded set ; but even if a space is not totally bounded, some of its subsets still will be.
* Every compact set is totally bounded, whenever the concept is defined.
* Every totally bounded metric space is bounded.
Every compact metric space is totally bounded.
* Every finite set of points is bounded
* Every relatively compact set in a topological vector space is bounded.
Every topological vector space X gives a bornology on X by defining a subset to be bounded iff for all open sets containing zero there exists a with.
Every ( bounded ) convex polytope is the image of a simplex, as every point is a convex combination of the ( finitely many ) vertices.

Every and linear
Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.
Every time a diode switches from on to off or vice versa, the configuration of the linear network changes.
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 smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:
* Every linear combination of its components Y = a < sub > 1 </ sub > X < sub > 1 </ sub > + … + a < sub > k </ sub > X < sub > k </ sub > is normally distributed.
Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
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 physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator.
Every vector in the space may be written as a linear combination of unit vectors.
Every nontrivial proper rotation in 3 dimensions fixes a unique 1-dimensional linear subspace of R < sup > 3 </ sup > which is called the axis of rotation ( this is Euler's rotation theorem ).
Every finite-dimensional normed space is reflexive, simply because in this case, the space, its dual and bidual all have the same linear dimension, hence the linear injection J from the definition is bijective, by the rank-nullity theorem.
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 real m-by-n matrix yields a linear map from R < sup > n </ sup > to R < sup > m </ sup >.
* Every ( biregular ) algebraic automorphism of a projective space is projective linear.
Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.
Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.
Every linear program has a dual problem with the same optimal solution, but the variables in the dual problem correspond to constraints in the primal problem and vice versa.
* Every irreducible polynomial in K which has a root in L factors into linear factors in L.
Every linear function on a finite-dimensional space is continuous.

Every and transformation
Every element in O ( 1, 3 ) can be written as the semidirect product of a proper, orthochronous transformation and an element of the discrete group
Every deck transformation permutes the elements of each fiber.
Every universal cover p: D → X is regular, with deck transformation group being isomorphic to the fundamental group.
Every Möbius transformation is a bijective conformal map of the Riemann sphere to itself.
Every six rounds, a logical transformation layer is applied: the so-called " FL-function " or its inverse.
Every mixing transformation is ergodic, but there are ergodic transformations which are not mixing.
Every orthogonal transformation of a k-frame in R < sup > n </ sup > results in another k-frame, and any two k-frames are related by some orthogonal transformation.
Every element x of G gives rise to a tensor-preserving self-conjugate natural transformation via multiplication by x on each representation, and hence one has a map.

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