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The product space , together with the canonical projections, can be characterized by the following universal property: if is a topological space, and for every ,: is a continuous map, then there exists precisely one continuous map : such that for each the following diagram commutes:
More abstractly, the outer product is the bilinear map (,) sending a vector and a covector to a rank 1 linear transformation (simple tensor of type (1, 1)), while the inner product is the bilinear evaluation map given by evaluating a covector on a vector; the order of the domain vector spaces here reflects the covector/vector distinction.
Another possibility is the product topology, where a base is also given by the Cartesian products of open sets in the component spaces, but only finitely many of which can be unequal to the entire component space. While the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties.
Given an H-space with multiplication :, the Pontryagin product on homology is defined by the following composition of maps (;) (;) (;) (;)where the first map is the cross product defined above and the second map is given by the multiplication of the H-space followed by application of the homology functor to obtain a homomorphism on the level of homology.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
A Customer Journey map is a visual or graphic interpretation of the overall story from an individual’s perspective of their relationship with an organization, service, product or brand, over ...
The tensor product of two vector spaces is a vector space that is defined up to an isomorphism.There are several equivalent ways to define it. Most consist of defining explicitly a vector space that is called a tensor product, and, generally, the equivalence proof results almost immediately from the basic properties of the vector spaces that are so defined.
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps.