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The sets of vectors representing two polytopes can be added by taking the union of the two sets and, when the two sets contain parallel vectors with the same sign, replacing them by their sum. The resulting operation on polytope shapes is called the Blaschke sum .
The sixteen dark-red points (on the right) form the Minkowski sum of the four non-convex sets (on the left), each of which consists of a pair of red points. Their convex hulls (shaded pink) contain plus-signs (+): The right plus-sign is the sum of the left plus-signs.
A subset of a vector space is called a cone if for all real >,.A cone is called pointed if it contains the origin. A cone is convex if and only if +. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones).
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .
When T is a continuous linear map between two topological vector spaces V and W, then the transpose T′ is continuous when W′ and V′ are equipped with "compatible" topologies: for example, when for X = V and X = W, both duals X′ have the strong topology β(X′, X) of uniform convergence on bounded sets of X, or both have the weak-∗ ...
If and are vector lattices with order complete and if is the set of all positive linear maps from into then the subspace := of (;) is an order complete vector lattice under its canonical order; furthermore, contains exactly those linear maps that map order intervals of into order intervals of . [5]
In mathematics, projections onto convex sets (POCS), sometimes known as the alternating projection method, is a method to find a point in the intersection of two closed convex sets. It is a very simple algorithm and has been rediscovered many times. [1] The simplest case, when the sets are affine spaces, was analyzed by John von Neumann.
In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution [1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category.