Inner Product Spaces
Definition (Inner Product Space)
An inner product (or scalar product) on a real or complex vector space is a mapping
that is
linear in its second argument
conjugate symmetric
nondegenerate
An inner product space (or pre-Hilbert space) is a pair consisting of a real or complex vector space and an inner product on .
Proposition (Norm Induced by an Inner Product)
If is an inner product on a real or complex vector space , then
defines a norm on .
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 and of a real inner product space
For all vectors and of a complex inner product space
Note that the complex polarization identity takes the slightly different form
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
for all vectors and . Taking the difference yields the desired result.
In the complex case, the inner product is conjugate symmetric, and we have
for all vectors and . This implies
The second equation follows from the first by substituting with and using that . To obtain the polarization Identity, multiply the second equation with and then add it to the first.
General Polarization Identity
Let be a complex inner product space. Let be a -th root of unity with and . Then
As a special case, for and , we obtain
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 be a real or complex normed space. A norm on is induced by an inner product on , if and only if satisfies the parallelogram law
In this case, the inner product is uniquely determined by and given by the polarization identity.
Stewart’s Theorem
Let , , be vectors of an inner product space. If , and are colinear and lies in between and , then we have
Cauchy–Schwarz Inequality
For all vectors and of an inner product space (with inner product and induced norm )
and equality holds precisely when and are linearly dependent.
Expressed only in terms of the inner product, the Cauchy–Schwarz Inequality reads
Proof TODO
Corollary (Continuity of the Inner Product)
The inner product is jointly norm continuous.