MATHEMATICS DEPARTMENT STANFORD UNIVERSITY
MATH 175 SPRING 2013 HOMEWORK 1
DUE AT LECTURE FRIDAY APRIL 12
1. For a sequence of real numbers x = (xi )N we dene for p > 0
1
p
x
p
|xi |p
:=
.
i=1
(a) (4 points) Show that p denes a norm for p = 1 and p = 2 (a
MATHEMATICS DEPARTMENT STANFORD UNIVERSITY
MATH 175 SPRING 2013 HOMEWORK 3
DUE AT LECTURE FRIDAY APRIL 26
1. Let U R be open. We consider the space L1 (U ) of functions f : U R satisfying
f
L1 (U )
|f (x)| dx < .
:=
U
(a) (3 points) Show that L1 (U ) is c
MATHEMATICS DEPARTMENT STANFORD UNIVERSITY
MATH 175 SPRING 2013 HOMEWORK 4
DUE AT LECTURE FRIDAY MAY 3
1. Consider the space of continuous functions C 0 ([1, 1]) with the norm f
Let U C 0 ([1, 1]) be the linear subspace
f C 0 ([1, 1]) |
f (t) dt = 0 .
f (
MATHEMATICS DEPARTMENT STANFORD UNIVERSITY
MATH 175 SPRING 2013 HOMEWORK 2
DUE AT LECTURE FRIDAY APRIL 19
1. We consider the function space
M := cfw_f : [1, 1] R | f smooth and f (1) = 0
and the inner products
1
1
f (t)g(t) dt
(f, g) =
(f, g)H0 =
1
and
1
MATHEMATICS DEPARTMENT STANFORD UNIVERSITY
MATH 175 SPRING 2013 HOMEWORK 7
DUE AT LECTURE FRIDAY MAY 31
1. Let V and W be Hilbert spaces, (ek )kN a complete orthonormal system of V and
(fk )kN an orthonormal system of W .
(a) (3 points) Assume that the se