Inner product space

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 $V$.

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 space of polynomials on $[0,1]$ with $\langle p,q\rangle=\int_0^1 p(t)q(t)\,dt$ is an inner product space.