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, $⟨x|y⟩={\sum }_{i=1}^n{x}_i{y}_i$