Inner product space

Definition

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

Remarks

A finite-dimensional real inner product space is a Euclidean space. Each inner product space has the induced norm $\norm{x} = \sqrt{\langle x, x \rangle}$. An inner product space that is complete for this norm is a Hilbert space. Each finite-dimensional inner product space is complete.

Explanations

L1 human

An inner product space is a vector space where we can measure lengths and angles. The space $\R^n$ with the dot product is the main example. Much of plane geometry, such as the Pythagorean theorem, works in every inner product space.