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* × *X* → *X* and
· : *K* × *X* → *X*, 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* cos*θ* =
∑_{i}* X*_{i}*Y*_{i}.

* __Vector (cross) product__: (*X*_{x}, *X*_{y}, *X*_{z}) ×
(*Y*_{x},* Y*_{y},* Y*_{z})
= (*X*_{y}*Y*_{z}–*X*_{z}*Y*_{y}, *X*_{z}*Y*_{x}–*X*_{x}*Y*_{z},
*X*_{x}*Y*_{y}–*X*_{y}*Y*_{x}),
which can also be expressed in determinant form, or as (**X** × **Y**)_{i} = ε_{ijk} *X*_{j}*Y*_{k}.

@ __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*):= ∫ d*z* *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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