Vectors  

In General
* Versions: An arrow between points on Rn; An infinitesimal displacement in a manifold; An element of a vector space; A contravariant rank-1 tensor, i.e., one of type (1,0).
@ References: Weinreich 98 [geometrical]; Lesche et al AJP(92) [dual/covector]; Schmidt AJP(97)gq [covariant vs contravariant].

Vector Space > s.a. Bilinear Form; Flag; Hahn-Banach Theorem; Subspace.
$ Def: A set X with two operations +: X × X → X and · : K × X → X, where K is a field (usually R or C), satisfying some properties.
* New vector spaces out of old: > see Direct Sum (of R-modules), direct or tensor product.
* Relationships: A vector space structure is stronger than an affine structure; > s.a. affine structure.
@ References: Halmos 74.

Topological Vector Space > s.a. Banach Space; Fréchet Space; norm.
$ Def: A vector space X over a topological field K such that the two operations are continuous; > see MathWorld, Wikipedia.
* Examples: Any metric space or normed vector space, including Banach spaces and Hilbert spaces.
* Remark: On a real or complex vector space there is a unique, natural topology that makes it a topological vector space (by the Tychonoff theorem).
* Locally convex: A tvs which admits a topological base of convex sets; A real vector space with a (Hausdorff) topology generated by a family of seminorms.
@ References: Bourbaki 66; Kelley 76.

Vector Algebra in Euclidean Space
* Scalar (dot) product: X · Y = XY cos = i XiYi.
* Vector (cross) product: (Xx, Xy, Xz) × (Yx, Yy, Yz) = (XyYz–XzYy, XzYx–XxYz, XxYy–XyYx), which can also be expressed in determinant form, or as (X × Y)i = ijk XjYk.
@ References: Silagadze JPA(02) [7D generalization of vector product].

Inner Product > s.a. distance; formulations of quantum mechanics; Schwarz Inequality; types of symplectic structures.
$ Def: A map   ,  : V × V → R (C), where V is a vector space over R (C), which is bilinear (sesquilinear), Hermitian, and positive definite.
* Examples: For probability distributions, the information metric [@ Groisser & Murray dg/96]; For complex functions,

(A, B):= dz A*(z) B(z) F(z) ,   F(z) > 0  for all z .

* And other structure: Can always be used to define a norm by x:= x, x1/2, and thus a distance.
* Inner product space: A pair (V,   ,  ) as above; An example is any Hilbert space.
@ References: Dvurecenskij LMP(01) [criterion for completeness].

Related Concepts > see Bivector; Riesz Space [vector lattice].

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Send feedback and suggestions to bombelli at olemiss.edu – Modified 5 jul 2008