Search results for "vector space"
showing 10 items of 287 documents
Remarks on the semivariation of vector measures with respect to Banach spaces.
2007
Suppose that and . It is shown that any Lp(µ)-valued measure has finite L2(v)-semivariation with respect to the tensor norm for 1 ≤ p < ∞ and finite Lq(v)-semivariation with respect to the tensor norm whenever either q = 2 and 1 ≤ p ≤ 2 or q > max{p, 2}. However there exist measures with infinite Lq-semivariation with respect to the tensor norm for any 1 ≤ q < 2. It is also shown that the measure m (A) = χA has infinite Lq-semivariation with respect to the tensor norm if q < p.
The Ptolemy and Zbăganu constants of normed spaces
2010
Abstract In every inner product space H the Ptolemy inequality holds: the product of the diagonals of a quadrilateral is less than or equal to the sum of the products of the opposite sides. In other words, ‖ x − y ‖ ‖ z − w ‖ ≤ ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ for any points w , x , y , z in H . It is known that for each normed space ( X , ‖ ⋅ ‖ ) , there exists a constant C such that for any w , x , y , z ∈ X , we have ‖ x − y ‖ ‖ z − w ‖ ≤ C ( ‖ x − z ‖ ‖ y − w ‖ + ‖ z − y ‖ ‖ x − w ‖ ) . The smallest such C is called the Ptolemy constant of X and is denoted by C P ( X ) . We study the relationships between this constant and the geometry of the space X , and hence with metric fix…
On Banaschewski functions in lattices
1991
hold for all x, y ~ X. We call such a function z a Banaschewski function or a B-function on X. A lattice L is a B-lattice or antitonely complemented, if there is a B-function defined on the whole lattice L. For instance, Boolean lattices as well as orthocomplemented lattices are B-lattices. On the other hand, a B-lattice is not necessarily Boolean or orthocomplemented, although a distributive B-lattice is a Boolean lattice. It is shown later that a matroid (geometric) lattice is also a B-lattice. Naturally, our results include the lemma of Banaschewski [ 1, Lemma 4], by which the lattice of the subspaces of a vector space is a B-lattice. It should be emphasized that a B-function is supposed…
On Certain Metrizable Locally Convex Spaces
1986
Publisher Summary This chapter discusses on certain metrizable locally convex spaces. The linear spaces used are defined over the field IK of real or complex numbers. The word "space" will mean "Hausdorff locally convex space". This chapter presents a proposition which states if U be a neighborhood of the origin in a space E. If A is a barrel in E which is not a neighborhood of the origin and F is a closed subspace of finite codimension in E’ [σ(E’,E)], then U° ∩ F does not contain A° ∩ F. Suppose that U° ∩ F contain A° ∩ F. Then A° ∩ F is equicontinuous hence W is also equicontinuous. Since W° is contained in A, it follows that A is a neighborhood of the origin, a contradiction.
The complex of words and Nakaoka stability
2005
We give a new simple proof of the exactness of the complex of injective words and use it to prove Nakaoka's homology stability for symmetric groups. The methods are generalized to show acyclicity in low degrees for the complex of words in "general position". Hm(§ni1;Z) = Hm(§n;Z) for n=2 > m where §n denotes the permutation group of n elements. An elementary proof of this fact has not been available in the literature. In the first section the complex C⁄(m) of abelian groups is studied which in de- gree n is freely generated by injective words of length n. The alphabet consists of m letters. The complex C⁄(m) has the only non vanishing homology in degree m (Theorem 1). This is a result of F.…
K-theory of function rings
1990
AbstractThe ring R of continuous functions on a compact topological space Xwith values in R or C is considered. It is shown that the algebraic K-theory of such rings with coefficients in ZkZ, k any positive integer, agrees with the topological K-theory of the underlying space X with the same coefficient rings. The proof is based on the result that the map from Rδ (R with discrete topology) to R (R with compact-open topology) induces a natural isomorphism between the homologies with coefficients in ZkZ of the classifying spaces of the respective infinite general linear groups. Some remarks on the situation with X not compact are added.
Local dimensions of sliced measures and stability of packing dimensions of sections of sets
2004
Abstract Let m and n be integers with 0 R n to certain properties of plane sections of μ. This leads us to prove, among other things, that the lower local dimension of (n−m)-plane sections of μ is typically constant provided that the Hausdorff dimension of μ is greater than m. The analogous result holds for the upper local dimension if μ has finite t-energy for some t>m. We also give a sufficient condition for stability of packing dimensions of section of sets.
Normed vector spaces consisting of classes of convex sets
1965
New dimension indices for the characterization of the solvent-accessible surface
2001
Phase Fourier vector model for scale invariant three-dimensional image detection.
2009
A scale invariant 3D object detection method based on phase Fourier transform (PhFT) is addressed. Three-dimensionality is expressed in terms of range images. The PhFT of a range image gives information about the orientations of the surfaces in the 3D object. When the object is scaled, the PhFT becomes a distribution multiplied by a constant factor which is related to the scale factor. Then 3D scale invariant detection can be solved as illumination invariant detection process. Several correlation operations based on vector space representation are applied. Results show the tolerance of detection method to scale besides discrimination against false objects.