Standard Inner Product

Notation

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

• $⟨u,v⟩$
• $⟨u|v⟩$
• $u\cdot v$

Relation to Norm

$⟨\alpha |\alpha ⟩=||\alpha |{|}^2$

Properties of Standard Inner Product

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 $u=\overrightarrow{0}$

Implementations

Defined differently for the Field:

• Standard Inner Product for Real Numbers (Dot product)
• Standard Inner Product for Complex Numbers
• Standard Inner Product for Vector Spaces
• Standard Inner Product for Function Spaces
• Standard Inner Product for Matrix Spaces

Theorems

• Inner Products Define Angles Theorem
• Orthoganality Theorem
• Standard Inner Product Function
• Matrix Representation of Inner Product
• Inner Products Define a Norm
• Riesz Representation Theorem