Problem set 3 - Due 03/19/2012
Functional analysis - spring 2012
1) Show that if X is innite dimensional and K is one-to-one operator of K (X ) then I K cannot be in K (X ).
2) Let A B (X, Y ) and K K (X, Y ) where X and Y are B-spaces. If R(A) R(K ), sho
Problem set 2 - Due 03/05/2012
Functional analysis - spring 2012
1) Suppose A is a linear operator on a Hilbert space X such that D(A) = X and (x, Ay ) = (Ax, y ) for all x, y X .
Show that A B (X ).
2) Let cfw_Dk be a sequence of dense open subsets of a
Math Analysis 9 week test review
2.1-2.4, anything from P1-P7 and 1.2-1.6
2.1: Linear & quadratic functions
Formulas: rate of change, vertical form of a quadratic function, vertical free fall
a) writing linear functions given 2 points:
write an equation f
3.6 Mathematics of finance
Formulas you will need to know:
Key
t
Simple Interest (interest compounded annually): A = P(1 + r)
Compounded interest (when you receive interest k times a year): A = P(1 +
Compounded continuously: A = Pe
A = amt. of $ after t t
Math Analysis Ch. 1 Test review
Whats on the test:
1.2:
Finding Domain: (remember these are the x values)
*if there is not a radical or a fraction, your domain will be (-,)
*if there is a radical set it = 0 and solve, then decide what can and cant be in t
Problem set 10 - Due 05/14/2012
Functional analysis - spring 2012
1) If M is a closed subspace of a seperable normed vector space X , show that X/M is also seperable.
2) Show that the range of a compact operator is seperable.
3) For X a B-space, show that
Problem set 9 - Due 04/30/2012
Functional analysis - spring 2012
1) Let A be a closed linear operator on X such that (A )1 is compact for some (A). Show that e (A)
is empty.
2) LEt A be a closed operator on X such that 0 (A). Show that if = 0 then (A) i 1
Problem set 8 - Due 04/23/2012
Functional analysis - spring 2012
1) Suppose is an isolated point of (A), A B (X ) and f (z ) is analytic in a neighborhood of . If f (A) = 0
show that f (z ) has a zero at .
2) Suppose A B (X ) and (A) is contained in the h
Problem set 7 - Due 04/16/2012
Functional analysis - spring 2012
1) If M is a subspace of a B-space X , we recall that M has nite codimension if codimM = dim(X/M ) < .
Show by an example that in this case, M is not necessary closed.
2) X and Y are B-space
Problem set 6 - Due 04/09/2012
Functional analysis - spring 2012
1) Let A and B be operators in B (X ) that commute. Show that
r (AB ) r (A)r (B ),
r (A + B ) r (A) + r (B ).
Can you nd examples of strict inqualities.
Is the commutation hypothesis necessa
Problem set 5 - Due 04/02/2012
Functional analysis - spring 2012
1) Show that if there is a nonegative k such that Ak is compact then I A is Fredholm.
2) X and Y are B-space. Show that if A B (X, Y ) and A K (Y , X ), then A K (X, Y ).
3) Suppose X is a B
Problem set 4 - Due 03/26/2012
Functional analysis - spring 2012
1) Show that a linear functional A on a B-space is bounded if and only if N (A) is closed.
2) Let T B (X, Y ). Then T is compact if and only if [T ] B (X/N (T ), Y ) is compact.
3) Let X = l