---
tag: 01EK923SKP
kind: definition
name: Inner product space
msc: [15A63, 46C05]
defines: 0QVNCJXRY3
uses: [0VGSYBWHV2]
sources:
  - url: https://en.wikipedia.org/wiki/Inner_product_space
    locator: Definition
---

## 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|0NB0AWGJBW]] $\langle \cdot, \cdot \rangle$ on $V$.

## Remarks

A finite-dimensional real inner product space is a *Euclidean space*.
Each inner product space has the [[induced norm|0MFJJV97RJ]] $\norm{x} = \sqrt{\langle x, x \rangle}$.
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.