Chapter II.3

Vector spaces with a scalar product

[35/71%]

Section II.3.1Euclidean vector spaces[8/100%]

Subsection II.3.1.1Heuristic considerations and definitions

Subsection II.3.1.2Main metric concepts

Subsection II.3.1.3The process of orthogonalization

Subsection II.3.1.4Isomorphisms of Euclidean vector spaces

Subsection II.3.1.5Orthogonal bases and orthogonal matrices

Subsection II.3.1.6Symplectic spaces

Section II.3.2Hermitian vector spaces[3/33%]

Subsection II.3.2.1Hermitian forms

Subsection II.3.2.2Metric relations

Subsection II.3.2.3Orthogonality

Subsection II.3.2.4Unitary matrices

Subsection II.3.2.5Normed vector spaces

Section II.3.3Linear operators on spaces with a scalar product[13/92%]

Subsection II.3.3.1The relationship between linear operators and *-linear forms

Subsection II.3.3.2Types of linear operators

Subsection II.3.3.3The canonical form of Hermitian operators

Subsection II.3.3.4Reduction of a quadratic form to the principal axes

Subsection II.3.3.5Reduction of a pair of quadratic forms to canonical form

Subsection II.3.3.6The canonical form of isometries

Subsection II.3.3.7Normal operators

Subsection II.3.3.8Positive definite operators

Subsection II.3.3.9Polar decomposition

Section II.3.4Complexification and realification[4/50%]

Subsection II.3.4.1Complex structure

Subsection II.3.4.2Realization

Subsection II.3.4.3Complexification

Subsection II.3.4.4Complexification — realification — complexification

Section II.3.5Orthogonal polynomials[7/29%]

Subsection II.3.5.1The approximation problem

Subsection II.3.5.2Least squares method

Subsection II.3.5.3Linear systems and the method of least squares

Subsection II.3.5.4Trigonometric polynomials

Subsection II.3.5.5Note on self-adjoint operators

Subsection II.3.5.6Legendre polynomials

Subsection II.3.5.7Weighted orthogonalization

Subsection II.3.5.8Chebyshev polynomials

Subsection II.3.5.9Hermite polynomials