Standard Inner Product for Vector Spaces

Definition

1. With $V$ as a vector space over $\mathbb{F}$
2. $\forall \alpha ,\beta ,\gamma \in V$, $\forall c\in \mathbb{F}$
3. An inner product $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$ has properties
   1. $⟨\alpha +\beta |\gamma ⟩=⟨\alpha +\beta ⟩+⟨\beta +\gamma ⟩$
   2. $⟨c\alpha |\beta ⟩=c⟨a|\beta ⟩$
   3. $⟨\beta |\alpha ⟩=\overline{⟨\alpha |\beta ⟩}$ (Conjugate Symmetry)
   4. $⟨\alpha |\alpha ⟩>0$ if $\alpha \ne 0$

Example

With $\alpha =(x_1,\dots ,x_n)$ With $\beta =(y_1,\dots y_n)$ Over $\mathbb{C}⟹\alpha \cdot \beta ={\sum }_{j=1}^n{x}_j{\overline{y}}_j=⟨\alpha ,\beta ⟩$

Proof

Proving $⟨\alpha +\beta |\gamma ⟩=⟨\alpha |\gamma ⟩+⟨\beta |\gamma ⟩,\forall \alpha ,\beta ,\gamma \in {\mathbb{R}}^n$

1. Over $\mathbb{C}$, $⟨\alpha +\beta |\gamma ⟩=⟨(x_1,\dots ,x_n)+(y_1,\dots ,y_n)|(z_1,\dots ,z_n)⟩$
2. $=⟨(x_1+y_1,\dots ,x_n+y_n|(z_1,\dots ,z_n)⟩$
3. $={\sum }_{i=1}^n(x_i+y_i){\overline{z}}_i$
4. $={\sum }_{i=1}^n(x_i){\overline{z}}_i+{\sum }_{i=1}^n{x}_i{\overline{z}}_i$
5. $=⟨\alpha ,\gamma ⟩+⟨\beta ,\gamma ⟩$ … for the other 4 axioms

Theorems

• Basic Properties of Inner Products
• Inner Products Define a Norm
• Nice Inner Product
• Function that Recovers the Inner Product
• Matrix Representation of Inner Product