Inner product space - Knowlpedia

An inner product space is a vector space $V$ over $\mathbb{R}$ or $\mathbb{C}$ together with a specified inner product $\langle\cdot,\cdot\rangle$ on $V$ .

Remarks

Every inner product space is automatically a normed vector space via $\|v\|=\sqrt{\langle v,v\rangle}$ , and many constructions in linear algebra are organized by orthogonality. A complete inner product space is a Hilbert space.

Examples

$\mathbb{R}^n$ with the standard dot product is an inner product space.
$\mathbb{C}^n$ with the standard Hermitian product is an inner product space.
The real vector space of real polynomials on $[0,1]$ with $\langle p,q\rangle=\int_0^1 p(t)q(t)\,dt$ is an inner product space.

Over $\mathbb C$ , use $\langle p,q\rangle=\int_0^1 p(t)\overline{q(t)}\,dt$ . The polynomial space is not complete for this integral norm, so an inner product space need not be a Hilbert space.