Standard Inner Product

Notation

Standard inner product of vector $u,v$ denoted as:

• $⟨u,v⟩$
• $⟨u|v⟩$
• $u\cdot v$ Defined differently for the Field:
• Standard Inner Product for Real Numbers (Dot product)
• Standard Inner Product for Complex Numbers

Operation

$\overrightarrow{u}\cdot \overrightarrow{v}=|\overrightarrow{u}|\ast |\overrightarrow{v}|\ast \cos \theta$ $\overrightarrow{u}\cdot \overrightarrow{v}=u_1{v}_1+\cdots +u_n{v}_n$ $(x+y)\cdot w=x\cdot w+y\cdot w$

Length of Norm

The length of the normal $\mid \mid v\mid \mid =\sqrt{(v,v)}=\sqrt{v_1^2+\cdots +v_n^2}$ Note that the inner product is linear in the sense that $⟨u,av=bw⟩=a⟨u,v⟩+b⟨u,w⟩$

Properties of Standard Inner Product over ${\mathbb{R}}^n$

1. $⟨u,v⟩=⟨v,u⟩$
2. $⟨u,v_1+v_2⟩=⟨u,v_1⟩+⟨u,v_2⟩$
3. $⟨u,av⟩=a⟨v,u⟩$
4. $⟨u,u⟩\ge 0$ with equality if $u=\overrightarrow{0}$

Proofs

• Proving Standard Inner Function Property 2
• Proving Standard Inner Function Property 3

Theorems

• Inner Products Define Angles Theorem
• Orthoganality Theorem
• Standard Inner Product for Complex Numbers
• Standard Inner Product Function
• Riesz Representation Theorem
• Standard Inner Product for Real Numbers