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Page "Universal property" ¶ 18

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Let and C

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

Let M be a smooth manifold and let x be a point in M. Let I x be the ideal of all functions in C ∞( M ) vanishing at x, and let I x 2 be the set of functions of the form, where f i , g i ∈ I x .

Let M be a smooth manifold and let f ∈ C ∞( M ) be a smooth function.

Let C and D be categories.

Let a trapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD.

Let ( X i , f ij ) be an inverse system of objects and morphisms in a category C ( same definition as above ).

Let H be a Hilbert space, and let H * denote its dual space, consisting of all continuous linear functionals from H into the field R or C. If x is an element of H, then the function φ x , defined by

If V is a real vector space, then we replace V by its complexification V ⊗ R C and let g denote the induced bilinear form on V ⊗ R C. Let W be a maximal isotropic subspace, i. e. a maximal subspace of V such that g | W = 0.

Let ρ be the initial topology on X induced by C τ ( X ) or, equivalently, the topology generated by the basis of cozero sets in ( X, τ ).

Let J and C be categories with J a small index category and let C J be the corresponding functor category.

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Let F be an arbitrary functor from C to Set.

Let A, B, C and D be the hexes that make up a rhombus, with A and C being the non-touching pair.

Let F: J → C be a diagram of type J in a category C. A cone to F is an object N of C together with a family ψ X : N → F ( X ) of morphisms indexed by the objects of J, such that for every morphism f: X → Y in J, we have F ( f ) o ψ X

Let F be a diagram that picks out three objects X, Y, and Z in C, where the only non-identity morphisms are f: X → Z and g: Y → Z.

Let J be a directed poset ( considered as a small category by adding arrows i → j if and only if i ≤ j ) and let F: J op → C be a diagram.

Let and be

Let the open enemy to it be regarded as a Pandora with her box opened ; ;

Let every policeman and park guard keep his eye on John and Jane Doe, lest one piece of bread be placed undetected and one bird survive.

`` Let him be now ''!!

Let us assume that it would be possible for an enemy to create an aerosol of the causative agent of epidemic typhus ( Rickettsia prowazwki ) over City A and that a large number of cases of typhus fever resulted therefrom.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let p be the minimal polynomial for T, Af, where the Af, are distinct irreducible monic polynomials over F and the Af are positive integers.

Let Af be the null space of Af.

Let N be a linear operator on the vector space V.

Let T be a linear operator on the finite-dimensional vector space V over the field F.

Let V be a finite-dimensional vector space over an algebraically closed field F, e.g., the field of complex numbers.

Let N be a positive integer and let V be the space of all N times continuously differentiable functions F on the real line which satisfy the differential equation Af where Af are some fixed constants.

Let Q be a nonsingular quadric surface bearing reguli Af and Af, and let **zg be a Af curve of order K on Q.

Let us take a set of circumstances in which I happen to be interested on the legislative side and in which I think every one of us might naturally make such a statement.

Let the state of the stream leaving stage R be denoted by a vector Af and the operating variables of stage R by Af.

Let this be denoted by Af.

Let it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.

Let not your heart be troubled, neither let it be afraid ''.

The same God who called this world into being when He said: `` Let there be light ''!!

For those who put their trust in Him He still says every day again: `` Let there be light ''!!

Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.

Let her out, let her out -- that would be the solution, wouldn't it??

Let and category

Let X and Y be objects of a category D. The product of X and Y is an object X × Y together with two morphisms

Let C be a category and let J be a small index category.

Let I be a finite category and J be a small filtered category.

* Let Met c be the category of metric spaces with continuous functions for morphisms.

Let C be a category with zero object.

Let C be a semiadditive category, so a category having

Let C be a category.

Let ( M ,⊗, I ,,, ) be a monoidal category.

Let be the category of finite sets and bijections ( the collection of all finite sets, and invertible functions between them ).

Let be a category with some objects and.

Let X be a topological space, and let C be a category.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

The formal definition is as follows: Let C be a category and let

Let C be a category.

Let S be a subcategory of a category C. We say that S is a full subcategory of C if for each pair of objects X and Y of S

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