V×W

V×W denotes the Cartesian product of two sets V and W. It consists of all ordered pairs (v, w) with v in V and w in W and is the standard way to form a new object from two given ones.

If V and W are vector spaces over a field F, V×W inherits a natural vector space structure called the external direct product. Addition and scalar multiplication are defined componentwise: (v1, w1) + (v2, w2) = (v1 + v2, w1 + w2) and a·(v, w) = (a·v, a·w). The projections π_V: V×W → V and π_W: V×W → W are linear maps. This construction satisfies the universal property of the product in the category of vector spaces: for any vector space X with linear maps to V and to W, there exists a unique linear map X → V×W making the diagram commute.

V×W is canonically isomorphic to the direct sum V⊕W; for finite-dimensional spaces the distinction is a matter of convention, and one may often identify V×W with V⊕W. In particular, if V ≅ F^m and W ≅ F^n, then V×W ≅ F^{m+n}.

It is important to distinguish V×W from the tensor product V⊗W. The product captures simultaneous maps to V and W, while the tensor product encodes bilinear maps from V×W with a different universal property.

Notes: the concept extends similarly to other structures (groups, modules, topological spaces) by equipping the product with the corresponding componentwise operations, making V×W a product object in those categories as well.