Standard Inner Product for Complex Numbers

Definition

• With $V={\mathbb{C}}^n$
• Fixing $\alpha \in {\mathbb{C}}^n$
• Let the inner product $<\cdot |\cdot >:{\mathbb{C}}^n\times {\mathbb{C}}^n\to \mathbb{C}$ be the standard inner product
• $x=(x_1,\dots ,x_{n)}$
• $y=(y_1,\dots ,y_n)$
• Then, $⟨x|y⟩={\sum }_{i=1}^n{x}_i{\overline{y}}_i$
  • where ${\overline{y}}_i$ is the Complex Conjugate of $y_i$

Theorems

• Riesz Representation Theorem