Inner Product Spaces

An inner product space consists of a vector field $V$ over the field $F$ endowed with a function $⟨.,.⟩:V\times V\to F$ which satisfies the following properties: .

1. Conjugate Symmetry: $⟨a,b⟩=\overline{⟨b,a⟩}$
2. Linearity in the First Argument: $⟨\alpha a,b⟩=\alpha ⟨a,b⟩$ and $⟨a+c,b⟩=⟨a,b⟩+⟨c,b⟩$
3. Positive-Definiteness: $⟨a,a⟩\ge 0$, with equality holding iff $a=\underset{―}{0}$

These properties imply many important properties such as the Triangle Inequality and the Cauchy Schwarz Inequality. The mapping is also an example of a sesquilinear form.

1. In ${\mathbb{R}}^n$ we have $⟨v_1,v_2⟩=a_1{b}_1+a_2{b}_2+...+a_n{b}_n$
2. For ${\mathbb{C}}^n$ in order to preserve positive definiteness we take $⟨v_1,v_2⟩=a_1\overline{b_1}+a_2\overline{b_2}+...+a_n\overline{b_n}$