# Inner Product Spaces

Definition (Inner Product Space)

An

$⟨⋅,⋅⟩:X×X→K$inner product(orscalar product) on a real or complex vector space $X$ is a mappingthat is

linear in its second argument

$⟨x,y+z⟩=⟨x,y⟩+⟨x,z⟩⟨x,αy⟩=α⟨x,y⟩$conjugate symmetric

$⟨x,y⟩ =⟨x,y⟩$nondegenerate

An

inner product space(orpre-Hilbert space) is a pair $(X,⟨⋅,⋅⟩)$ consisting of a real or complex vector space $X$ and an inner product $⟨⋅,⋅⟩$ on $X$.

Proposition (Norm Induced by an Inner Product)

If $⟨⋅,⋅⟩$ is an inner product on a real or complex vector space $X$, then

$∥x∥=⟨x,x⟩ ∀x∈X$defines a norm on $X$.

In this sense, every inner product space is also a normed space. As a consequence it is also a metric space and a topological space.

The next theorem shows how the inner product can be recovered from the norm.

Polarization Identity

For all vectors $x$ and $y$ of a real inner product space

$4⟨x,y⟩=∥x+y∥_{2}−∥x−y∥_{2}.$For all vectors $x$ and $y$ of a complex inner product space

$4⟨x,y⟩=∥x+y∥_{2}−∥x−y∥_{2}+i∥x−iy∥_{2}−i∥x+iy∥_{2}.$

Note that the complex polarization identity takes the slightly different form

$4⟨x,y⟩=∥x+y∥_{2}−∥x−y∥_{2}+i∥x+iy∥_{2}−i∥x−iy∥_{2},$if we follow the convention that the inner product is conjugate linear in its second argument.

Proof In the real case, the inner product is symmetric, and we have

$∥x±y∥_{2}=∥x∥_{2}±2⟨x,y⟩+∥y∥_{2}$for all vectors $x$ and $y$. Taking the difference yields the desired result.

In the complex case, the inner product is conjugate symmetric, and we have

$∥x±y∥_{2}=∥x∥_{2}±2Re⟨x,y⟩+∥y∥_{2}$for all vectors $x$ and $y$. This implies

$∥x+iy∥_{2}−∥x−iy∥_{2}∥x−iy∥_{2}−∥x+iy∥_{2} =4Re⟨x,y⟩,=4Im⟨x,y⟩. $The second equation follows from the first by substituting $y$ with $−iy$ and using that $Re⟨x,−iy⟩=Re(−i⟨x,y⟩)=Im⟨x,y⟩$. To obtain the polarization Identity, multiply the second equation with $i$ and then add it to the first. $□$

General Polarization Identity

Let $X$ be a complex inner product space. Let $ζ$ be a $n$-th root of unity with $ζ=1$ and $ζ_{2}=1$. Then

$⟨x,y⟩=n1 k=0∑n−1 ζ_{k}∥x+ζ_{k}y∥_{2}∀x,y∈X.$

As a special case, for $ζ=i$ and $n=4$, we obtain

$⟨x,y⟩=41 k=0∑3 i_{k}∥x+i_{k}y∥_{2}.$Proof TODO $□$

For an arbitrary normed space, the polarization identity does not, in general, define an inner product. The following theorem, gives a condition for when it does.

Parallelogram Law

Let $X$ be a real or complex normed space. A norm $∥⋅∥$ on $X$ is induced by an inner product $⟨⋅,⋅⟩$ on $X$, if and only if $∥⋅∥$ satisfies the

$∥x+y∥_{2}+∥x−y∥_{2}=2∥x∥_{2}+2∥y∥_{2}∀x,y∈X.$parallelogram lawIn this case, the inner product is uniquely determined by $∥⋅∥$ and given by the polarization identity.

Stewart’s Theorem

Let $x$, $y$, $z$ be vectors of an inner product space. If $x$, $y$ and $z$ are colinear and $y$ lies in between $x$ and $y$, then we have

$∥p−x∥_{2}∥y−z∥+∥p−z∥_{2}∥x−y∥=(∥p−y∥_{2}+∥x−y∥∥y−z∥)∥x−z∥$

Cauchy–Schwarz Inequality

For all vectors $x$ and $y$ of an inner product space (with inner product $⟨⋅,⋅⟩$ and induced norm $∥⋅∥$)

$∣⟨x,y⟩∣≤∥x∥∥y∥,$and equality holds precisely when $x$ and $y$ are linearly dependent.

Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads

$∣⟨x,y⟩∣_{2}≤⟨x,x⟩⟨y,y⟩.$Proof TODO $□$

Corollary (Continuity of the Inner Product)

The inner product is jointly norm continuous.