Let T : C([0, 1]) → R be a (not necessarily bounded) linear functional.
Show that T is positive if and only if 
= 
(here 1 denotes the constant function [0, 1] → R, x → 1).
Homework Answers
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Let T : C([0, 1]) → R be a (not necessarily bounded) linear
functional.
Show that...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R
be continuous, and suppose that |f(x)| ≤ g(x) for all x ∈ U. Show
that if
exists, then so does
.
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Q. Determine whether the given functions are exponentially bounded and piecewise continuous on 0 ≤ t...
Q. Determine whether the given functions are exponentially
bounded and piecewise continuous on 0 ≤ t < ∞.
(a) f(t) = tant
(b) f(t) = cosh2t
(c) f(t) =
, where
denotes the greatest integer less than or equal to t.
We were unable to transcribe this imageWe were unable to transcribe this image
Let T: V V and S: V V and R: V V be three linear operators...
Let T: V
V and S: V
V and R: V
V be three linear operators on V. Suppose we have
T
S= S
R , Then prove ker(S) is an invariant subspace for R .
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10. Mobius transformations. Let a, b, c, d ad-bc 0 . The function is called a...
10. Mobius transformations. Let
a, b, c, d ad-bc 0 . The
function is called a Mobius transformation (or linear fractional
transformation). Show that
a) lim z->inf T(z) = inf if c=0;
b)kim z-> inf T(z) = a/c and lim z-> d/c T(z) = inf if
c0
*10. Möbius transformations. Let a,b,c,d EC with ad-bc70. The function T(2) = 2 a2 + b cz + d à (2 +-d/c) is called a Möbius transformation (or linear fractional transformation). Show that...
Let n, and let n be a reduced residue. Let r = odd(). Prove that if...
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
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Let E be the solid bounded by the planes , , , , . Set up...
Let E be the solid bounded by the planes , , , , . Set up all six orders of integration for the evaluation of as an iterated integral. We were unable to transcribe this imageWe were unable to transcribe this imagey=0 We were unable to transcribe this imageWe were unable to transcribe this imagef(x, y, 2)d
Let ⊂ be a rectangle and let f be a function which is integrable on R....
Let ⊂
be a
rectangle and let f be a function which is integrable on R. Prove
that the graph of f, G(f) := {(x, f(x)) ∈
: x ∈ }, is a
Jordan region and that it has volume 0 (as a subset of
).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X(t) = 2; if 0 t 1; 3; if 1 t 3; -5; if 3 t 4: or in one formula X(t) = 2I[0;1](t) + 3I...
Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Use the transformation and to evaluate the integral where is the region bounded on the by...
Use the transformation
and
to evaluate the integral
where
is the region bounded on the
by the ellipse
Let S be the image of R under
T on the .
Sketch regions
R and S. Set up the integral as
an iterated integral of a function
over region S. Use technology to evaluate the
integral. Give the exact answer.
We were unable to transcribe this imageWe were unable to transcribe this imageR xdA We were unable to transcribe this imageWe were...
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space t...
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space to itself, and is an eigenvalue of , then is a simple eigenvalue of if . 1. Suppose is a vector space of dimension over field where you may assume that is either or , and let be a linear map from to . Show...
Let U ⊆ R^n be open (not necessarily bounded), let f, g : U → R
be continuous, and suppose that |f(x)| ≤ g(x) for all x ∈ U. Show
that if
exists, then so does
.
We were unable to transcribe this imageWe were unable to transcribe this image
Q. Determine whether the given functions are exponentially
bounded and piecewise continuous on 0 ≤ t < ∞.
(a) f(t) = tant
(b) f(t) = cosh2t
(c) f(t) =
, where
denotes the greatest integer less than or equal to t.
We were unable to transcribe this imageWe were unable to transcribe this image
Let T: V
V and S: V
V and R: V
V be three linear operators on V. Suppose we have
T
S= S
R , Then prove ker(S) is an invariant subspace for R .
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
10. Mobius transformations. Let
a, b, c, d ad-bc 0 . The
function is called a Mobius transformation (or linear fractional
transformation). Show that
a) lim z->inf T(z) = inf if c=0;
b)kim z-> inf T(z) = a/c and lim z-> d/c T(z) = inf if
c0
*10. Möbius transformations. Let a,b,c,d EC with ad-bc70. The function T(2) = 2 a2 + b cz + d à (2 +-d/c) is called a Möbius transformation (or linear fractional transformation). Show that...
Let n,
and let
n
be a reduced residue. Let r = odd().
Prove that if r = st for positive integers s and t, then
old(t)
= s.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let ⊂
be a
rectangle and let f be a function which is integrable on R. Prove
that the graph of f, G(f) := {(x, f(x)) ∈
: x ∈ }, is a
Jordan region and that it has volume 0 (as a subset of
).
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let X(t) =
2; if 0 t 1;
3; if 1 t 3;
-5; if 3 t 4:
or in one formula X(t) = 2I[0;1](t) +
3I(1;3](t) -
5I(3;4](t).
Give the Itˆo integral
X(t)dB(t)
as a sum of random variables, give its distribution,
specify the mean and the variance.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe...
Use the transformation
and
to evaluate the integral
where
is the region bounded on the
by the ellipse
Let S be the image of R under
T on the .
Sketch regions
R and S. Set up the integral as
an iterated integral of a function
over region S. Use technology to evaluate the
integral. Give the exact answer.
We were unable to transcribe this imageWe were unable to transcribe this imageR xdA We were unable to transcribe this imageWe were...
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