Inner Products and Norms

What is an inner product on a vector space V?

An inner product is a function \langle.,.\rangle:V\times V\rightarrow\mathbb{F} satisfying for all u,v,w\in V, \alpha\in\mathbb{F}:

1. Linearity in the first argument

2. Conjugate symmetry: \langle v,u\rangle =\overline{\langle u,v\rangle}

3. Positive-definiteness: \langle vv\rangle \geq0 with equality iff v=0.

What is a real inner product?

An inner produce over \mathbb{R} satisfying:

• \langle u,v\rangle=\langle v,u\rangle (symmetry)

• \langle u,u\rangle >0 for u\neq 0

What is the standard inner product on \mathbb{F}^n?

For u=(u_1,…,u_n) and v=(v_1,…,v_n), \langle u,v\rangle=\sum^n_{i=1}u_i\overline{v_i}. This is the usual dot product in \mathbb{R}^n and Hermitian form in \mathbb{C}^n.

What is the norm of a vector in an inner product space?

\|v\|=\sqrt{\langle v,v\rangle}. It measures the “length”" of the vector.

What is the distance between two vectors in an inner product space?

dist(u,v)=\|u-v\|