Inner Product Space

linear-logic

Definition

Inner Product Space

Let $V$ be a vector space over a field $F$. An inner product space is a pair $⟨V,⟨\cdot ,\cdot ⟩⟩$, where $⟨\cdot ,\cdot ⟩:V\times V\to F$ is a map, called an inner product, that satisfies the following axioms for all vectors $x,y,z\in V$ and scalars $\alpha ,\beta \in F$:

• Conjugate Symmetry:

$$⟨x,y⟩=\overline{⟨y,x⟩}$$

• Linearity in the first argument:

$$⟨\alpha x+\beta y,z⟩=\alpha ⟨x,z⟩+\beta ⟨y,z⟩$$

• Positive Definiteness:

$$\begin{array}{rl}⟨x,x⟩& \ge 0\\ ⟨x,x⟩& =0⟺x=0\end{array}$$

Induced Norm

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

$$||x|:=\sqrt{⟨x,x⟩}$$