15 - Inner Product Spaces

Inner Product Space

Let $V$ be a real vector space then

$V$ with a bilinear, symmetric positive definite form $F(\cdot ,\cdot )$, more commonly written as

$$⟨\cdot ,\cdot ⟩$$

Complex Inner Product Space

Let $V$ be a complex vector space then

$V$ with a sesquilinear, conjugate symmetric, positive definite form $F(\cdot ,\cdot )$, more commonly written as

$$⟨\cdot ,\cdot ⟩$$

Mutually Orthogonal Set

Let $V$ be a complex or real inner product space

$\{w_1,\cdots ,w_n\}$ is mutually orthogonal if

$$⟨w_i,w_j⟩=0\text{ for all }i\ne j$$

Orthonormal Set

Let $V$ be a complex or real inner product space

$\{w_1,\cdots ,w_n\}$ is orthonormal if they are mutually orthogonal and

$$⟨w_i,w_i⟩=1\text{ for all }i$$

Linearly Independence of an Orthogonal Set

Let $V$ be an inner product over $\mathbb{K}$ (equal $R$ or $C$)
Let $\{w_1,\cdots ,w_n\}\subset V$ be orthogonal with $w_i\ne 0$ for all $i$ then

$$w_1,\cdots ,w_n\text{ are linearly independent }$$

Proof

Assume ${\sum }_i{\lambda }_i{w}_i=0$ for some ${\lambda }_i\in \mathbb{K}$ then
For all $j$ then

$$⟨w_j,\sum _i{\lambda }_i{w}_i⟩=0$$

However

$$⟨w_j,\sum _i{\lambda }_i{w}_i⟩=\sum _i{\lambda }_i⟨w_j,w_i⟩={\lambda }_j⟨w_j,w_j⟩$$

Hence ${\lambda }_j=0$ for all $j$

Existence of Orthonormal Bases corollary

Every finite dimensional inner product space $V$ over $\mathbb{K}=\mathbb{R},\mathbb{C}$ has an orthonormal basis

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Length Determines Inner Product

The length function determines the inner product

Given two inner products $⟨,{⟩}_1$ and $⟨,{⟩}_2$ then

$$⟨v,v{⟩}_1=⟨v,v{⟩}_2\forall v\in V⟺⟨v,w{⟩}_1=⟨v,w{⟩}_2\forall v,w\in V$$

Proof

Implication for $⟸$ is trivial
For $⟹$ note that

$$⟨v+w,v+w⟩=⟨v,v⟩+⟨v,w⟩+\overline{⟨v,w⟩}+⟨w,w⟩$$

For $\mathbb{K}=\mathbb{C}$ then

$$⟨v+iw,v+iw⟩=⟨v,v⟩+i⟨v,w⟩-i\overline{⟨v,w⟩}+⟨w,w⟩$$

Hence

$$\begin{array}{rlrl}\mathrm{Re}⟨v,w⟩& =& & \frac{1}{2}(||v+w|{|}^2-||v|{|}^2-||w|{|}^2)\\ \mathrm{Im}⟨v,w⟩& =-& & \frac{1}{2}(||v+iw|{|}^2-||v|{|}^2-||w|{|}^2)\end{array}$$

Hence the inner product is given in terms of the length function