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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 C be the category of vector spaces K-Vect over a field K and let D be the category of algebras K-Alg over K ( assumed to be unital and associative ).

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 algebraic

Let V 1 and V 2 be algebraic varieties.

Let V be an algebraic set defined as the set of the common zeros of an ideal I in a polynomial ring over a field K, and let A = R / I be the algebra of the polynomials over V. Then the dimension of V is:

Let be any set, let be an algebraic structure of type generated by.

Let the underlying set of this algebraic structure, sometimes called universe, be, and let be a function.

Let P ( x, y ) and Q ( x, y ) be the polynomials defining the algebraic curves we are interested in.

Let R be the ring of integers of an algebraic number field K and P a prime ideal of R. For each extension field L of K we can consider the integral closure S of R in L and the ideal PS of S. This may or may not be prime, but assuming is finite it is a product of prime ideals

Let K be a field and L a finite extension ( and hence an algebraic extension ) of K. Multiplication by α, an element of L, is a K-linear transformation

Let ƒ: X → Y be a morphism of algebraic curves.

* Let us use the term “ deductive system ” as a set of sentences closed under consequence ( for defining notion of consequence, let us use e. g. Tarski's algebraic approach ).

Let be a variety defined over the finite field with elements and let be the lift of to the algebraic closure of.

Let G be a semisimple Lie group or algebraic group over, and fix a maximal torus T along with a Borel subgroup B which contains T. Let λ be an integral weight of T ; λ defines in a natural way a one-dimensional representation C λ of B, by pulling back the representation on T = B / U, where U is the unipotent radical of B.

Let V be an affine algebraic variety of dimension d defined by a prime ideal I =⟨ f 1 , ..., f k ⟩ in.

Let P4 be the space of all the algebraic polynomials in degree less than 4 ( i. e. the highest exponent of x can be 3 ).

* Let K a be an algebraic closure of K containing L. Every embedding σ of L in K a which restricts to the identity on K, satisfies σ ( L ) = L. In other words, σ is an automorphism of L over K.

Let X be a complex algebraic manifold and a good compactification.

Let K be an algebraic number field K. Its Dedekind zeta function is first defined for complex numbers s with real part Re ( s ) > 1 by the Dirichlet series

Let D be the vector space of rational divisor classes on V, up to algebraic equivalence.

Let X be an affine algebraic variety embedded into the affine space k n , with the defining ideal I ⊂ k. For any polynomial f, let in ( f ) be the homogeneous component of f of the lowest degree, the initial term of f, and let in ( I ) ⊂ k be the homogeneous ideal which is formed by the initial terms in ( f ) for all f ∈ I, the initial ideal of I.

Let X be an algebraic variety, x a point of X, and ( O X, x , m ) be the local ring of X at x.

The minimal separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an ( a, b )- separator S can be regarded as a predecessor of another ( a, b )- separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two ( a, b )- separators in ' G '.

Let k be an algebraic number field with ring of integers and let be a prime ideal.

Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V. The set Hom ( Y, X ) of morphisms of algebraic spaces is then defined by the condition that it makes the descent sequence

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