Vectors and Vector Spaces  

In General > s.a. vector field.
* Versions: An arrow between points on \(\mathbb R\)n; An infinitesimal displacement in a manifold; An element of a vector space; A contravariant rank-1 tensor, i.e., one of type (1, 0).
* In physics: Example of the types of vectors used in physics are Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra.
@ References: Weinreich 98 [geometrical]; Lesche et al AJP(92)jun [dual/covector]; Fleisch 11 [II]; Chappel et al IEEE(16)-a1509 [different vector formalisms, historical perspective].
@ Covariant and contravariant: Schmidt AJP(97)nov-gq; Kumar a1002 [pedagogical].

Vector Space > s.a. Bilinear Form; Flag; Hahn-Banach Theorem; Subspace.
$ Def: A set X with two binary operations +: X × XX and  · : K × XX, where K is a field (usually \(\mathbb R\) or \(\mathbb C\)), satisfying some properties.
* Linearly independent vectors: A set of two or more vectors in a vector space is linearly independent if the only linear combination of the vectors in the set that equals 0 is the one with all coefficients equal to zero; Alternatively, none of the vectors is a linear combination of the others.
* New vector spaces out of old: > see Direct Sum (of R-modules), direct or tensor product.
* Ordered vector space: A vector space with a partial order < satisfying simple compatibility conditions with the addition and scalar multiplication.
* Relationships: A vector space structure is stronger than an affine structure; > s.a. affine structure.
@ References: Halmos 74.
> Online resources: see MathWorld page; Wikipedia page.

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.
* 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.
> Online resources: see MathWorld page; Wikipedia page.

Vector Algebra in Euclidean Space
* Scalar (dot) product: X · Y = XY cosθ = ∑i XiYi.
* Vector (cross) product: (Xx, Xy, Xz) × (Yx, Yy, Yz) = (XyYzXzYy, XzYxXxYz, XxYyXyYx), which can also be expressed in determinant form, or as (X × Y)i = εijk XjYk.
@ References: Silagadze JPA(02) [7D generalization of vector product].

Inner Product > s.a. distance; formulations of quantum mechanics; hilbert space; Schwarz Inequality; types of symplectic structures.
$ Def: A map \(\langle\ ,\  \rangle: V\times V\to {\mathbb R}\ ({\mathbb C})\), where V is a vector space over \(\mathbb R\) (\(\mathbb C\)), 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||:= \(\langle\)x, x\(\rangle\)1/2, and thus a distance.
* Inner product space: A pair (V, \(\langle\) ,  \(\rangle\)) as above; An example is any Hilbert space.
@ General references: Dvurečenskij LMP(01) [criterion for completeness]; Horváth JGP(10) [semi-definite and indefinite inner products, generalized Minkowski spaces].
@ Partial inner product spaces: Antoine et al AMP-a1203 [categorical aspects]; Antoine & Trapani JPA(13)-a1210.

Related Concepts > see Bivector; Riesz Space [vector lattice].
* Axial vector or pseudovector: A vector whose definition implies that the vector corresponding to a mirror reflected situation is the opposite of the mirror image of the original vector; > s.a. Wikipedia page.


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