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Let be the dual space of.
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Let and be
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 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 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 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 it be granted then that the theological differences in this area between Protestants and Roman Catholics appear to be irreconcilable.
Let us therefore put first things first, and make sure of preserving the human race at whatever the temporary price may be ''.
Let and dual
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 < sub > x </ sub >, defined by
Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.
Let V be a vector space over a field k. The core of the contraction operation, and the simplest case, is the natural pairing of V with its dual vector space V *.
Let V < sup >∗</ sup > be the dual vector space of V. In other words, V < sup >∗</ sup > is the set of all linear maps from V to C with addition defined over it in the usual linear way, but scalar multiplication defined over it such that for any z in C, ω in V < sup >∗</ sup > and X in V. This is usually rewritten as a contraction with a sesquilinear form 〈·,·〉.
Let Λ be the lattice in R < sup > d </ sup > consisting of points with integer coordinates ; Λ is the character group, or Pontryagin dual, of R < sup > d </ sup >.
Precisely, let T be a triangulation of an n-manifold M. Let S be a simplex of T. We denote the dual cell ( to be defined precisely ) corresponding to S by DS.
Let G be a locally compact abelian group and G < sup >^</ sup > be the Pontryagin dual of G. The Fourier-Plancherel transform defined by
Let E < sub > i </ sub > be a basis of sections of TG consisting of left-invariant vector fields, and θ < sup > j </ sup > be the dual basis of sections of T < sup >*</ sup > G such that θ < sup > j </ sup >( E < sub > i </ sub >) = δ < sub > i </ sub >< sup > j </ sup >, the Kronecker delta.
Let B be an arbitrary Banach space, and let B < sup >*</ sup > be its dual, that is, the space of bounded linear functionals on B.
Let and space
Let M be a smooth manifold and let x be a point in M. Let T < sub > x </ sub > M be the tangent space at x.
Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.
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