Vectors and Vector Spaces |

**In General** > s.a. vector calculus; 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__:
It 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.
stochastic quantization [axial vector gauge theory];
Wikipedia page.

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