Inner Product Space

linear-logic

Definition

Inner Product Space

An inner product space $⟨V,⟨\cdot ,\cdot ⟩⟩$ is a vector space $V$ with an inner product $⟨\cdot ,\cdot ⟩:V\times V\to \mathbb{F}$.

Induced Norm

Every inner product space is naturally a normed vector space. The norm is defined “for free” using the inner product:

$$||x|:=\sqrt{⟨x,x⟩}$$