Standard Inner Product for Real Numbers

This is not a operation you can always do.

Definition

• With $V={\mathbb{R}}^n$
• Fix $\alpha \in {\mathbb{R}}^n$
• The inner product $⟨\cdot |\cdot ⟩:V\times V\to \mathbb{R}$
• With $x=(x_1,\dots ,x_n)$
• With $y=(y_1,\dots ,y_n)$ Then, $⟨x|y⟩={\sum }_{i=1}^n{x}_i{y}_i$ Alternatively, $⟨x|y⟩=x^{T}y$

Alternate Definitions

$⟨\overrightarrow{u}|\overrightarrow{v}⟩=|\overrightarrow{u}|\ast |\overrightarrow{v}|\ast \cos \theta$ $⟨\overrightarrow{u}|\overrightarrow{v}⟩=u_1{v}_1+\cdots +u_n{v}_n$

Proofs

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