Inner product space
Definition
An inner product space is a kind of vector space that has an additional structure called an inner product. This inner product allows lengths and angles to be defined. The inner product space is a key concept in functional analysis and used in many areas of mathematics, physics, and engineering.
Structure
An inner product space is a vector space V over the field F which is either the complex numbers C or the real numbers R, together with an inner product, i.e., with a map
- ⟨⋅,⋅⟩: V × V → F
that satisfies the following three properties for all vectors x, y, z in V and all scalars α in F:
1. Conjugate symmetry:
- ⟨x,y⟩ = ⟨y,x⟩*
where the star denotes the complex conjugate.
2. Linearity in the first argument:
- ⟨αx + y,z⟩ = α⟨x,z⟩ + ⟨y,z⟩
3. Positive-definiteness:
- ⟨x,x⟩ ≥ 0 with equality if and only if x = 0.
The first argument is said to be linear, or antilinear, if for all scalars α and β and all vectors x and y, the following equation holds:
- ⟨αx + βy,z⟩ = α⟨x,z⟩ + β⟨y,z⟩
Normed Vector Space
Every inner product space is a normed vector space, a space with a measure of vector length or magnitude. This is achieved by defining the norm of a vector x as the square root of the inner product of the vector with itself:
- ‖x‖ = √⟨x,x⟩
This norm satisfies the properties of a norm and defines a metric, or a measure of distance, between two vectors.
Orthogonality
In an inner product space, the concept of orthogonality is defined using the inner product. Two vectors x and y are orthogonal if their inner product is zero. This concept generalizes the idea of perpendicular vectors in Euclidean space to higher dimensions and to complex vector spaces.
Hilbert Spaces
An important type of inner product space is a Hilbert space, which is a complete inner product space. Completeness refers to the property that every Cauchy sequence in the space has a limit in the space. Hilbert spaces are fundamental to the mathematical formulation of quantum mechanics.
Applications
Inner product spaces are used in many areas of mathematics and its applications. In physics, the state space of quantum mechanics is a complex Hilbert space. In statistics, the method of least squares leads to a problem of minimizing a quadratic function, which can be solved using the inner product structure of Euclidean space.
