Last updated: 4 November 2014
An inner product space is a vector space over with a function such that
(a) | If and then |
(b) | If and then |
(c) | If then |
(d) | If and then |
(e) | If then |
Let be an inner product space. Define by
HW: Show that is a normed vector space.
A Hilbert space is an inner product space such that is a complete metric space.
Let be an inner product space and let be a subspace of
The orthogonal complement of in is
If is a Hilbert space and is closed then
Let be an inner product space and let be a subspace of An orthogonal projection onto is a linear transformation such that
(a) | if then |
(b) | if then |
(sub-Theorem) Let be a Hilbert space and let be a subspace of There exists an orthogonal projection onto if and only if is closed.
HW: Let be an inner product space. Show that is a linear transformation. Let be a subspace of Show that
Let be a Hilbert space.
An orthonormal sequence in is a sequence in such that if then
HW: Let be an orthonormal sequence in Let Show that
(a) | If then |
(b) | given by is an orthogonal projection onto |
(c) | If then is a Schauder basis of i.e., every can be written uniquely as |
HW: (Fourier analysis) Let in be given by Show that is an orthonormal basis of
Let be a sequence of linearly independent vectors in Define Then is an orthonormal sequence in
These are a typed copy of Lecture 33 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 23, 2014.