B4b Hilbert spaces HT 2013: Sheet 4 (Hilbert spaces and the Projection Theoremcorrected)
Problem 4.1. Let H be a Hilbert space and let C be a non-empty closed convex subset of H. Recall from
Lecture 8 that for each x X there is a unique y = PC (x) C such
Introduction to Complex Numbers
Problem Sheet
Michaelmas 2015
Exercise 1 Which of the following quadratic equations require the use of complex numbers to solve
them?
3x2 + 2x 1 = 0, 2x2 6x + 9 = 0, 4x2 + 7x 9 = 0.
Exercise 2 Put each of the following comp
Running head: QDT1 ABSTRACT ALGEBRA TASK 3
QDT1 Abstract Algebra Task 3
Student's Name
Institution Affiliation
1
QDT1 ABSTRACT ALGEBRA TASK 3
2
QDT1 Abstract Algebra Task 3
A. Let G be the set of the fifth roots of unity.
1. Use de Moivres formula to veri
13.1 Compound Interest
Simple interest interest is paid only on the
principal
Compound interest interest is paid on both
principal and interest, compounded at regular
intervals
Example: a $1000 principal paying 10% simple
interest after 3 years pays .1
B4b Hilbert spaces HT 2013: Problem Sheet 2 (OMT CGT and strong convergence)
Problem 2.1 (Strong operator convergence). This question revisits some ideas from Week 1 lectures. Let X and Y
be Banach spaces and let (Tn )nN be a sequence of bounded linear op
B4b Hilbert spaces HT 2013: Problem Sheet 3(Inner products and Hilbert spaces)
Problem 3.1. Let X be a complex vector space. Then X can be considered also as a real vector space XR , simply
by restricting scalar multiplication to real scalars.
(a) Let , b
B4b Hilbert spaces HT 2012: Solutions to Sheet 1
Problem 1.1. Let M be a complete metric space and, for each n N, let En be a nowhere dense subset of M and Gn
be a dense open subset of M . Show that nN Gn is not contained in nN En . Deduce that Q is not t
B4b Hilbert spaces HT 2013: Sheet 7 (Self-adjointness and orthonormality.)
Problem 7.1. Let T be a self-adjoint operator on a Hilbert space H. Show that if T is surjective then T is
invertible. [As well as ideas from recent lectures, one of the theorems p
B4B HILBERT SPACES: EXTENDED SYNOPSES
Lecture 1: Historical remarks about Hilbert, his 23 problems, his spaces, relevance to
integral equations, quantum theory, Fourier series. The diculty of giving an example
of a discontinuous linear operator between Ba
B4b Hilbert spaces HT 2013: Sheet 6 (Adjoint operators.)
Problem 6.1.
(1) Let H be a Hilbert space and let P be a bounded linear projection on H. (Recall that
saying P is a projection means that P 2 = P .) Prove that
P = P (im P ) = ker P
P 1.
(2) Show t
B4b Hilbert spaces HT 2012: Problem Sheet 2 (OMT CGT and strong convergence)
Problem 2.1 (Strong operator convergence). This question revisits some ideas from Week 1 lectures. Let X and Y
be Banach spaces and let (Tn )nN be a sequence of bounded linear op
B4b Hilbert spaces HT 2013: Problem Sheet 1 (Baires Theorem and Uniform Boundedness)
Problem 1.1. Let M be a complete metric space and, for each n N, let En be a nowhere dense subset of M
and Gn be a dense open subset of M . Show that n=1 Gn is not contai
B4b Hilbert spaces HT 2013: Problem Sheet 3(Inner products and Hilbert spaces)
Problem 3.1. Let X be a complex vector space. Then X can be considered also as a real vector space XR , simply
by restricting scalar multiplication to real scalars.
(a) Let , b
Introduction to Complex Numbers
Frances Kirwan, based on notes by Balzs Szendri and Richard Earl
a
o
Michaelmas 2013
The shortest path between two truths in the real domain passes through the complex domain.
Jacques Hadamard (1865-1963)
1
Complex numbers: