![]() ![]() Let a set S be given in ℝ n, and the sets Γ, U, Γ j ( d ), S j ( d ) and U j ( d ), constructed in the previous sections, correspond to it. It is also true for the conical and convex hulls. ![]() Here linear subspaces and manifolds are mapped into linear subspaces and manifolds with the dimension preserved. The affine transformations (11.1) and (11.2) are one-to-one mappings of the spaces ℝ * n and ℝ n into themselves. these spaces are conjugate (or dual) to each other. Thus, the scalar product (11.3) is preserved with the transformations (11.1) and (11.2) in ℝ * n and ℝ n respectively, i.e. But this follows from the fact that the m linearly independent (over the reals) vectors e 1,…, e m, are linear combinations of the f i (since M is a lattice, we need only verify that the vectors f 1,…, f l are linearly independent over the real numbers. M * must possess a basis of l ≤ m vectors f 1,…, f l. Applying Theorem 2 of Section 2, we see that the subgroup M, which means that x is a linear combination, with integer coefficients, of (1/ j) e 1,…, (1/ j) e m. Since the order of any element of the factor group M is finite, which means that the index ( This shows that the number of vectors u which occur in (3.11) for all x ∈ M is discrete, T can contain only a finite number of vectors of ![]() M 0, constructed with the basis e 1,…, e m. M 0 and u lies in the fundamental parallelepiped T of the lattice ![]()
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