Vector spaces

Vector space V over field F

Vector space is closed under linear combination of a set of ‘basis’ vectors: A commutative group wrt +. Linear dependence of vectors: Any of the vectors expressible as a linear combo of the others.

Basis sets of n-degree polynomials ( $P_{n}$ ) and matrices also define vector spaces.

Inner product space

A vector space V with an inner product $⟨ . ⟩ : V \times V \to F$ .

Normed vector space

Space with a norm. Also a metric space. Thence inherit notion of completeness.

Lebesgue space

Aka $L^{p}$ or $l^{p}$ space. Infinite dimensional space with the p norm. (Minkowski) Triangle inequality still holds.

Banach space

Complete, normed, vector space.

Hilbert space

Hilbert noticed common theme: complete, normed, inner-product vector space.

Complex vector space

C is a field; so multiplication is defined for complex numbers. So, a complex vector space $C^{n}$ is not equivalent to thinking about real vector space $R^{2 n}$ .

Functional space V

f() as dom(f) dim vector

Look upon f(x) as a vector whose dimensions = domain size d. A dimension for each value of $x$ in a certain interval: $f (x_{1})$ is the projection of f(x) along the $x_{1}$ direction.

Infinite dimensions

Dimension of the function space could be $\infty$ or it could be finite depending on domain of f: see boolean functions ref.

Standard basis functionals

Let basis function/ direction along $x_{i}$ be $e_{i}$ : then by usual notion of inner product, $e_{i} ⊥ e_{j}$ .

By geometric intuition, tringle inequality, cosine inequality hold. So, have a inner product vector space!

Restriction to finite length

Consider functionals f(x) which are of finite length, even if you are in an $\infty$ dimensional vector space : Eg: $f (x) = \sin x$ in $[0, 2 π]$ , not $x^{- 1}$ in $[0, 2 π]$ . Otherwise, hard to make sense of triangle inequality.

Other representations

The space of all polynomials can be represented both as a functional space described above, and as a vector space, where each polynomial is represented by the vector formed by its coefficients.

Euclidian space

$R^{n}$ with the Euclidian structure (metric, inner product): $⟨ a, b ⟩ = \sum a_{i} b_{i}$ .

Geometric properties

For geometric properties of various objects in 3-d euclidian space, see topology ref.

Orthant: a generalization of quadrant.

Box measure

Aka Lebesgue measure. This is the minimum cover measure described in the algebra survey.

There exist sets which are not box-measurable!\why

Definition

For boxes, this is just the product measure: $m ([a, b]) = \prod_{i = 1}^{n} (b_{i} - a_{i})$ .

Let $B_{i} (S)$ be a set of disjoint boxes which cover $S$ . In general, $m (S) = inf {B_{i} (S)}$ .

Properties

It has all properties of a measure. In addition, observe that $m ([a, a]) = 0$ . So, the measure of any countable set of points is $0$ . So, measure of rationals $m (Q) = 0$ , whereas $m (R - Q) = 1$ .

Dual vector space $V^{*}$

Vector space over F of all continuous linear (not affine) functionals: $V \to F$ , with addition op: $(f + g) (x) = f (x) + g (x)$ .

Linear functionals $f (x)$ can always be specified as $f^{T} x$ .

If V has inner product, $V^{*}$ has inner product.

Dual of a dual space includes the original space \chk.

This concept finds important applicaitons: Eg: Dual cone, dual norms.

Basis: $\set{e^{i}$

${e^{i} : e^{i} (e_{j}) = 1 iff j = i}$ . For the finite dimensional case, this is simply another finite dimensional vector space.