### Vector Space Definitions

#### Definition of a Normed Vector Space

A set of objects (vectors) $\{\vec{u}, \vec{v}, \vec{w}, \dots\}$ is said to form a linear vector space over the field of scalars $\{\lambda, \mu,\dots\}$ (e.g. real numbers or complex numbers) if:

1. the set is closed, commutative, and associative under (vector) addition;
2. the set is closed, associative, and distributive under multiplication by a scalar;
3. there exists a null vector $\vec{0}$;
4. multiplication by the scalar identity $1$ leaves the vector unchanged;
5. all vectors have a corresponding negative vector;

#### Definition of an Inner Product

An inner product $\left\langle \vec{u}\vert \vec{v}\right\rangle$ is a generalization of the dot product with the following properties:

$$\left\langle \vec{u}\vert \vec{v}\right\rangle = \left\langle \vec{v}\vert \vec{u}\right\rangle^*$$

$$\left\langle \vec{u}\vert \lambda\vec{v}+\mu\vec{w}\right\rangle=\lambda\left\langle \vec{u}\vert \vec{v}\right\rangle+\mu\left\langle \vec{u}\vert \vec{w}\right\rangle$$