Math 7330: Functional Analysis Notes In the following H is a complex Hilbert space.
Fall 2002 A. Sengupta
1
Orthogonal Projections
We shall study orthogonal projections onto closed subspaces of H . In summary, we show: If X is any closed subspace of H the
Math 115a Lecture 3 Winter 2009
Midterm 1
Name:
Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise, you may use without proof anything proven in the sections of the book covered by this test (excluding the
MATH 23b, SPRING 2003 THEORETICAL LINEAR ALGEBRA AND MULTIVARIABLE CALCULUS Midterm Solutions (in-class portion) March 19, 2003
1. True or False T or F Every bounded innite set in Rn has an accumulation point. True. This is the Bolzano-Weierstrass Theorem
Math 115a Lecture 3 Winter 2009
Final Exam
Name:
Instructions: There are 8 problems. Make sure you are not missing any pages. Unless stated otherwise, you may use without proof anything proven in the sections of the book covered by this test (excluding th
Math 7330: Functional Analysis Notes 7: The Spectral Theorem
Fall 2002 A. Sengupta
Let B be a complex, commutative B algebra, with its Gelfand spectrum. Then, as we have seen in class, (i) the Gelfand transform B C () : x x satises x = x for every x B ; (
Math 7330: Functional Analysis Notes/Homework 6: Banach *-Algebras
Fall 2002 A. Sengupta
An involution on a complex algebra B is a map : B B for which (i) (a + b) = a + b for all a, b B (ii) (a) = a for all C and a B (iii) (xy ) = y x for all x, y B (iv)
Math 7330: Functional Analysis Homework 5: Commutative Banach Algebras II
Fall 2002 A. Sengupta
We work with a complex commutative Banach algebra B . It had been shown that the set G(B ) of all invertible elements in B is an open subset of B . A proper id
Math 7330: Functional Analysis Notes/Homework 4: Commutative Banach Algebras I
Fall 2002 A. Sengupta
1. Let R be a commutative ring with multiplicative identity e. A subset S R is an ideal of R if :(a) 0 S , (b) x + y S for every x, y S , and (c) rx S for
Math 7330: Functional Analysis Notes/Homework 3: Banach Algebras
Fall 2002 A. Sengupta
A complex algebra is a complex vector space B on which there is a bilinear multiplication map B B B : (x, y ) xy which is associative. Bilinearity of multiplication mea
Math 7330: Functional Analysis Homework 2
Fall 2002 A. Sengupta
In the following, H is a complex Hilbert space with a Hermitian inner-product (, ). All operators are operators on H . 1. Suppose P and Q are orthogonal projections. (i) Show that if P Q = QP
Math 7330: Functional Analysis Homework 1
Fall 2002 A. Sengupta
In the following, V is a nite-dimensional complex vector space with a Hermitian inner-product (, ), and A : V V a linear map. 1. Let e1 , ., en be an orthonormal basis of V . (i) Show that th
Ribets Math 110 Second Midterm, problems and abbreviated solutions Please put away all books, calculators, and other portable electronic devicesanything with an ON/OFF switch. You may refer to a single 2-sided sheet of notes. When you answer questions, wr