256 Inner Product Space

A set of vectors equipped with an inner product that is used to define distance and length of vectors from that inner product.

definition [D] (Inner Product Space) A vector space equipped with an inner product that associates a complex scalar with every ordered pair of vectors, where the following conditions apply:

The mapping satisfies conjugate symmetry: \(\langle \alpha | \beta \rangle = \langle \beta | \alpha \rangle^*\).
The mapping is linear in the second factor: \(\langle \alpha | a\beta + b\gamma \rangle = a\langle \alpha | \beta \rangle + b\langle \alpha | \gamma \rangle\).
The mapping is positive definite: \(\langle \alpha | \alpha \rangle \ge 0\), vanishing only if \(\alpha = 0\).

where

\(V\) is a vector space over a field \(K\).
\(\langle \alpha | \beta \rangle\) is the inner product of vectors \(\alpha\) and \(\beta\).
\(K\) is the scalar field.
\(\|\alpha\| = \sqrt{\langle \alpha | \alpha \rangle}\) is the norm.

Note:

a vector space is also called a linear space.
the inner product is also called the scalar product.
\(K\) is typically \(\mathbb{R}\).
\(K\) is also typically \(\mathbb{C}\).
the norm generalizes the concept of length.

256.1 Elementary Example

256.1.1 Simple

An inner product space is a vector space with an inner product. Take \(\mathbb{R}^{2}\) with the dot product.

\[ V = \mathbb{R}^{2} \]

\[ \langle u,v \rangle = u_{1} v_{1} + u_{2} v_{2} \]

\[ \lVert u \rVert = \sqrt{\langle u,u \rangle} \]

where

\(V\) is the inner product space.
\(\lVert u \rVert\) is the induced norm.

256.1.2 General

The space \(\mathbb{C}^{3}\) with the Hermitian inner product is an inner product space.

\[ V = \mathbb{C}^{3} \]

\[ \langle u,v \rangle = \sum_{j=1}^{3} u_{j} \overline{v_{j}} \]

where

distance is defined by \(d(u,v) = \lVert u - v \rVert\).

255 Inner Product

257 Kronecker Delta