Advanced Linear Algebra, Lecture 5.9: Complex inner product spaces

Advanced Linear Algebra, Lecture 5.9: Complex inner product spaces

Roughly speaking, complex inner product spaces are just like real inner product spaces, except that the "transpose" operation is replaced by "conjugate transpose". Formally, this is done by replacing the bilinear form with a "sesquilinear" form (sesqui- means "one and a half"). This is not a different definition, but rather, a way to extend the real case to the complex numbers. In other works, when restricting to the real numbers, everything remains bilinear. We discuss what adjoints and orthogonality means in a complex inner product space, and how the concept of an "orthogonal map" extends to a "unitary map". We conclude with an example of how complex Fourier series, using complex exponentials, can be formalized via a complex inner product structure, and how this compares to the real Fourier series that use sine and cosine functions, which have previously seen.

Course webpage: http://www.math.clemson.edu/~macaule/...