1.4 Inner product spaces

Where normed vector spaces incorporate the concept of length into a vector space, inner product spaces incorporate the concept of angle.

Definition 1

• IP1. $⟨x,y⟩=\overline{⟨y,x⟩}$
• IP2. $⟨\alpha x,y⟩=\alpha ⟨x,y⟩$
• IP3. $⟨x+y,z⟩=⟨x,z⟩+⟨y,z⟩$
• IP4. $⟨x,x⟩\ge 0$ with equality iff $x=0$ .

A vector space together with an inner product is called an inner product space .

Note that a valid inner product space induces a normed vector space with norm $∥x∥=\sqrt{⟨x,x⟩}$ . (Proof relies on Cauchy-Schwartz inequality.) In ${\mathbb{R}}^N$ or ${\mathbb{C}}^N$ , the standard inner product induces the ${\ell }_2$ -norm. We summarize the relationships between the various spaces introduced over the last few lectures in [link] .