# Inner Product Spaces

## Table of Contents

## 1 Definition

An **inner product space** consists of a vector field \( V \) over the field \( F \) endowed with a function \( \langle . \ , \ . \rangle : V \times V \to F \) which satisfies the following properties:
.

**Conjugate Symmetry**: \( \langle a, \ b \rangle = \overline{\langle b, \ a \rangle} \)**Linearity in the First Argument**: \( \langle \alpha a, \ b \rangle = \alpha \langle a, \ b \rangle \) and \( \langle a + c, \ b \rangle = \langle a, \ b \rangle + \langle c, \ b \rangle \)**Positive-Definiteness**: \( \langle a, \ a \rangle \ge 0 \), with equality holding iff \( a = \underline{0} \)

These properties imply many important properties such as the **Triangle Inequality** and the **Cauchy Schwarz Inequality**. The mapping is also an example of a **sesquilinear form**.

## 2 Examples

- In \( \mathbb{R}^n \) we have \( \langle v_1, \ v_2 \rangle = a_1b_1 + a_2b_2 + ... + a_nb_n \)
- For \( \mathbb{C}^n \) in order to preserve positive definiteness we take \( \langle v_1, \ v_2 \rangle = a_1\overline{b_1} + a_2\overline{b_2} + ... + a_n\overline{b_n} \)