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 \(\dprod{.}:V\times V \to F\).
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 \(R^{2n}\).
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, \(e_{i} \perp e_{j}\).
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 \([0,2\pi]\), not \(x^{-1}\) in \([0,2\pi]\). 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): \(\dprod{a, b} = \sum a_{i}b_{i}\).
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, \(m(S) = \inf \set{B_i(S)}\).
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: \(\set{e^{i}\)
\(\set{e^{i}: e^{i}(e_j) = 1 \texttt{iff j = i}}\). For the finite dimensional case, this is simply another finite dimensional vector space.